Inequality, Portfolio Choice and the Business Cycle

Heejeong Kim∗

Concordia University

Abstract

I study the effect of heterogeneity, in both the level and composition of wealth, in a

dynamic stochastic overlapping-generations economy where households face uninsurable

unemployment, earnings, and liquidity risk. Households pay transactions costs when they

adjust savings held in high expected return assets, which make such savings illiquid. They

also hold liquid, lower return assets. I show that household-level disparities in liquidity are

important for understanding differences in their behavior, as well as aggregate changes in

consumption and investment, over the Great Recession. When I allow for a rise in both

unemployment and disaster risk, reducing households’ expected income and the expected

return on their illiquid savings, aggregate consumption and investment fall to levels seen

in the recession.

The response of aggregate consumption is sensitive to the behavior of wealth poor

households with a high marginal utility of consumption. Facing a large possible fall in

earnings, they build precautionary savings in liquid assets. However, in a typical incomplete

markets model with a single asset, all households would respond to the fall in the expected

return on savings following a rise in disaster risk. The resulting substitution effect would

offset much of the negative wealth effect on aggregate consumption. In contrast, when

much of wealth is illiquid, many households do not respond to a fall in its expected return,

substantially dampening the substitution effect. Moreover, wealthier households, more

likely to adjust their portfolios, increase their shares of liquid assets. This results in a large

fall in aggregate investment.

Keywords: Business cycle, disaster risk, portfolio choice, wealth inequality

∗ Department of Economics, Concordia University, CA. E-mail: heejeong.kim.macro@gmail.com. I thank

Aubhik Khan and Jonathan Heathcote for their invaluable comments. I also thank Mariacristina De Nardi ,

Gianluca Violante, Fabrizio Perri, Andrew Glover, Pok-sang Lam, and Jesus Fernandez-Villaverde. I thank

session participants at the 2016 EuropeanWinter Meeting of the Econometric Society, 2017 North American

Summer Meeting, 2017 Midwest Macro Meeting and workshop participants at Ohio State University for

comments and suggestions that have improved this paper.

1 Introduction

Models of the business cycle have typically assumed a single asset held by households.

This includes recent heterogeneous agent models studying the effects of wealth inequality

on the propagation of large recessions (Krueger et al., 2016). However, the data shows large

differences in the composition of household wealth. Households not only make consumption-

savings decisions but also choose how to allocate savings across assets of varying liquidity

and returns. I quantify the importance of this channel for understanding business cycles and

differences in their impact on rich and poor households. While these are natural questions

to explore, the difficulty of solving a heterogeneous household model with multiple assets

in dynamic stochastic general equilibrium has meant that there has been little work done

in developing an answer.

Households’ portfolios of assets vary systematically with their age and wealth.1 In a

large recession, reductions in earnings and increases in unemployment risk may lead to a

reduction in the share of illiquid but productive assets. When accompanied by a rise in

precautionary savings, this may help explain large declines in aggregate investment and

consumption as well as a slow recovery. An evaluation of the quantitative importance

of this channel is the goal of this paper. In particular, I examine how optimal portfolio

choices, across liquid and illiquid assets that vary in their rates of return, affect consumption

and savings of different households. Importantly, this heterogeneity in the composition of

wealth, previously disregarded in business cycle models with uninsurable earnings risk, has

a quantitatively important role in reproducing the large declines in aggregate quantities

seen in the Great Recession.

While heterogeneity in the composition of wealth has recently received attention in

quantitative macroeconomic models, it has either been studied in endowment economies

(Glover et al., 2016) or in partial equilibrium (Kaplan and Violante, 2014), making it dif-

ficult to quantitatively evaluate the importance of this channel for the business cycle.

Moreover, in the absence of intra-generational inequality, it is hard to study the effects of

changes in household portfolios on the cross-sectional distributions of net worth, income

1See Glover et al., 2016 and Khan and Kim, 2015.

1

and consumption (Glover et al., 2016). Households of different ages hold different shares

of illiquid wealth, which implies dissimilar impacts of asset price changes on their income

and consumption. These differences are equally evident when examining households of

different wealth. Thus, a realistic assessment of the effect of portfolio choice requires intra-

generational differences that have been omitted in general equilibrium studies. To my

knowledge, I am the first to explore a quantitative DSGE OLG framework where house-

holds, who face earnings, unemployment and liquidity risk, choose both their consumption

and savings in low-yield liquid and high-yield illiquid assets. Liquidity risk arises through

transaction costs of actively adjusting illiquid assets. It reinforces the response of house-

holds to a rise in earnings and unemployment risk.

Another contribution of this paper is to introduce disaster risk into a heterogeneous

household economy and explore the role of a rise in the risk of economic disaster in the

presence of multiple assets of varying liquidity. The rise in disaster risk lowers households’

expected income and wealth. Studying the micro-data using a simple permanent income

model, De Nardi et al. (2011) argue that negative wealth effects, which might be driven by

the fall in the expected asset prices and the expected income, are crucial determinants for

understanding the fall in aggregate consumption during the Great Recession. In my model,

a rise in disaster risk, alongside a fall in total factor productivity, generates a strong response

in precautionary savings by all but the wealthiest households. This leads to a shift into

liquid assets. Important in this mechanism, which matches well household level data over

the Great Recession, is the assumption that economic disasters, when they happen, involve

a relatively large fall in TFP which reduces households’ expected life-time earnings and

the expected return to illiquid assets. This is in contrast to the existing implementation

of disaster risk taken by Gourio (2012), who studies its asset pricing implications in a

representative agent model. In the current work, there is a large recession when, following

a fall in TFP, a rise in disaster risk makes households increase precautionary savings and

reduce illiquid, productive assets.

Calibrating the model economy using household-level data, the distribution of transac-

tions costs that gives rise to illiquidity in capital investment is quantitatively disciplined

by reproducing the share of illiquid assets to net worth across households of different levels

2

of wealth and age. In deciding their savings in liquid and illiquid assets, which vary in

their usefulness for consumption smoothing, households face borrowing limits that are a

common percentage of age-specific natural debt limits that arise naturally in a life-cycle

model with retirement. These age-varying borrowing limits are crucial in reconciling the

micro-evidence showing a high share of illiquid assets for the young and wealth poor. The

model reproduces a share of liquid assets of about one-third, similar to the SCF. Addi-

tionally, allowing for heterogeneity in the return on savings across households reproduces a

significant fraction of the distributions of net worth, liquid assets, and illiquid assets seen

in the data.

The economy, with assets that vary in their liquidity and return, exhibits an aggregate

response to shocks that varies with the cyclicality of the return to liquid assets supplied

by the government. When this supply is weakly countercyclical, the model can exhibit

unusually large declines in investment and consumption over a recession. The steep fall in

earnings, and rise in unemployment risk, that characterizes a recession compels households

to smooth consumption by monetizing their illiquid assets. This substitution of liquid

for illiquid assets leads to an unusually large decline in investment when compared to a

single-asset economy.

A rise in the risk of a large economic disaster, during a recession, amplifies the effect of

the fall in productivity and leads to larger declines in aggregate consumption and invest-

ment. The heightened risk of a further worsening in earnings exacerbates an already large

negative wealth effect, driving a rise in precautionary savings and reducing consumption.

Investment also plummets as the increased disaster risk lowers the expected return on cap-

ital. This results in a large portfolio adjustments into liquid assets, especially when the

supply of liquid assets expands to maintain a constant safe real interest rate. When the

government finances a constant return on liquid savings, the wealth composition hetero-

geneity model with a rise in disaster risk predicts peak-to-trough declines of 4.1 percent in

consumption and 18 percent in investment, similar to the changes observed over the Great

Recession. This is the result of a rise in the risk of a disaster that interacts with multiple

assets of varying liquidity.

Importantly, in a single-asset economy without illiquid assets, a rise in disaster risk has

3

little effect as increases in precautionary savings’ motives are largely offset by equilibrium

changes in the return to capital. In the present work, households increase their precaution-

ary savings and decrease their consumption in a large recession when they expect lower

life-time earnings following a rise in disaster risk. (wealth effect) However, in a single as-

set economy, all households respond to a fall in the return to savings which dampens the

fall in consumption. (substitution effect) Thus, the substitution effect offsets the negative

wealth effect, mitigating the effect of disaster risk. In contrast, when households hold both

liquid and illiquid assets as in my model, a large fraction of non-adjusting households do

not respond to a fall in the return to capital (illiquid assets). This sharply decreases the

magnitude of the substitution effect, resulting in a large fall in aggregate consumption

consistent with the data.

Heterogeneity in the composition of wealth, and the associated countercyclical substi-

tution of liquid, low-yield assets for illiquid, productive investment in capital, leads to a

slow recovery following a large recession. The half-life of aggregate consumption rises 1.4

times compared to that in a single asset economy. As the economy begins to recover, house-

holds that paid transactions costs to reduce their holdings of illiquid assets and smooth

consumption against a fall in income are initially reluctant to reinvest in capital. Their

shares of illiquid assets, while less than desired, remain in a range consistent with optimal

adjustment in response to fixed costs. These households’ tolerance for portfolio imbalances

slows aggregate investment in capital and economic recovery.

Finally, I evaluate the models’ predictions for changes in the cross-sectional distribution

of net worth, income, and consumption to those in the PSID. During ordinary times, the

portfolio choice economy successfully explains growth rates of net worth, earnings, income,

and consumption that fall in wealth. These growth rates fall in a recession. More impor-

tantly, with countercyclical supply of liquid assets, savings rates rise in a large recession.

In a typical model recession, spending rates would rise as households smooth their con-

sumption. In contrast, household level data shows a fall in spending rates over the recent

recession. My model economy suggests an importance of the interplay, between a rise in

the probability of a large disaster and multiple assets of varying liquidity, for understand-

ing this fall in spending rates in the Great Recession. Thus, the model with heterogeneity

4

in the liquidity, as well as the level of wealth, and uninsurable earnings, employment and

liquidity risk, is consistent with an unusual characteristics of the household data over the

recession.

The numerical method developed to solve the model may be of independent interest.

As pointed out in Kaplan and Violante (2014), solving for a stochastic OLG economy with

a bivariate cross-sectional distribution of assets is challenging using the Krusell-Smith algo-

rithm as this approach involves a repeated long simulations to obtain an accurate parametric

law of motion for the aggregate state. First, I manage to overcome this difficulty by using

a two-stage approach to solve decision rules, which allows me to use the Endogenous grid

method to solve for the choice of liquid assets. Next, I extend the Backwards Induction

method of Reiter (2002, 2010) to solve a stochastic life-cycle model with heterogeneity in

household wealth, multiple assets, borrowing limits that vary in age, and fixed transactions

costs. In contrast to linearization methods, it is important to note that the Backward

Induction method allows aggregate risk to matter for household decisions and aggregate

dynamics.2

The remainder of the paper is organized as follows. Section 2 discusses the related

literature. Section 3 documents the cross-sectional distribution of net worth, earnings,

income, and consumption from the PSID. Section 4 presents the model economy. Section 5

discusses the calibration. Section 6 presents quantitative results and Section 7 concludes.

2 Related Literature

This paper contributes to the literature that studies the impact of a recession on het-

erogeneous households. Krueger et al. (2016) is the first to study changes in the joint

distribution of earnings, income, and consumption during the Great Recession in a hetero-

geneous households framework. They also explore the importance of cross-sectional wealth

inequality for the aggregate dynamics of an incomplete-markets Krusell-Smith economy

with unemployment and earnings risk as well as permanent differences in households’ dis-

count factors. They find that an economy with a more pronounced dispersion of wealth

2For linearization methods in continuous time, see Winberry et al., 2016 and Kaplan et al., 2016.

5

experiences a larger drop in consumption compared to an economy with less inequality

as wealth-poor households sharply decrease their consumption in a recession. I further

study changes in the joint distribution during the Great Recession by allowing differences

in households not just at the level of wealth but also in the composition of their wealth. In

addition, I explore how this endogenous portfolio channel alongside a rise in disaster risk

amplifies the Great Recession.

Guerrieri and Lorenzoni (2015) study the effects of a credit crunch during a recession

in a heterogeneous-agent incomplete markets model with variable labor supply and id-

iosyncratic income shocks. They find that a credit crunch forces financially constrained

households to repay their debt while it increases precautionary savings by unconstrained

households, resulting in a drop in real interest rates. As financially unconstrained house-

holds respond to a fall in real interest rates by increasing their consumption, the resulting

aggregate output and consumption responses are modest.3 In their extended model with

durable goods, output even increases by 0.4 percent following a credit crunch due to the

high substitutability between bonds and durables. In contrast, liquidity risk embedded

in illiquid wealth weakens substitutability between two assets in my model, generating a

larger drop in consumption in a recession.

My work is also related to a body of work that studies the effects of multiple assets

on heterogeneous households’ behavior. Glover et al. (2016) study the intergenerational

redistribution effects of the Great Recession in a stochastic complete-markets 6-period

OLG economy with i.i.d aggregate shocks. Studying financial markets characterized by

two assets - a risk-free bond and a risky equity, they find that older cohorts are hurt

more than younger cohorts as a result of a larger decline in asset prices compared to a

decline in wages during the Great Recession. However, in the absence of cross-sectional

heterogeneity in each cohort, they are unable to address effects of wealth inequality in

the recession. Moreover, given fixed total stocks of capital and labor, all changes in the

demand translate to security prices changes. On the contrary, wealth inequality arising

from financial friction with rich heterogeneity across households, in my model, makes it

3In their benchmark economy, a 10 percent drop in debt-to-GDP ratio only leads to one percent dropin output.

6

possible to explore the impact of the Great Recession across households with different

levels of wealth. Furthermore, my model economy possibly explain both price and quantity

effects of the asset demand during the Great Recession by allowing the aggregate capital

to change.

Kaplan et al. (2015) explore the consumption response to changes in house prices over

the Great Recession. Their OLG economy has two assets, housing and liquid savings.

However, by assuming an exogenous interest rate on liquid savings and exogenous labor

supply, which is the only factor of production, the dynamics of consumption, investment,

and GDP over the business cycle cannot be addressed by their framework.

My model economy is closely related to the framework of Kaplan and Violante (2014)

which studies the consumption responses to fiscal stimulus in an incomplete-markets life-

cycle model with an optimal portfolio choice between a low-return liquid asset and a high-

return illiquid asset involving a deterministic transaction cost to adjust a high-return illiquid

asset. In contrast to a partial equilibrium analysis in their work, I solve the model in a

general equilibrium to study implications of multiple assets that vary by liquidity and

risk over the business cycle. In addition, I explore the effects of a disaster risk affecting

households’ belief during the Great Recession.

3 Impact of the Great Recession on heterogeneous

households

Understanding changes in the joint distribution of net worth, consumption, earnings,

and income may be important for studying the Great Recession. In this section, using

the 2005-2011 PSID data, I document changes in the cross-sectional distributions of these

variables before and during the Great Recession.45 Disaggregated wealth data is contained

in the PSID since 2003, consisting of transaction accounts, stocks, bonds, IRAs, business

and farm equity, and debt. Consistent with the definition of net worth in the SCF, I define

4Krueger et al. (2016) also examined changes in the joint distribution of income, consumption, and networth during the recent recession.

5The PSID consists of the SRC and SEO samples. Though SEO sample is designed to oversample thepoor, I only used SRC sample to be consistent with sample selection for earnings estimation.

7

net worth as the sum of total assets minus total debt.6 Table 1 compares the distribution

of wealth in the PSID to that in the SCF. Although the PSID constructed measures of

wealth are less dispersed than those in the SCF data, it still explains around two-thirds of

total wealth held by the top 1 percent of household as well as the wealth Gini of 0.72 in

2007. Recent waves of the PSID also provide detailed spending data. I aggregate this data

to measure total expenditure, defined as the sum of total spending on nondurable goods

and services.7

Table 1. The distribution of wealth

Year Gini top 1% 5% 10% 50% 90% ≤ 02007 SCF 0.78 29.1 52.3 64.3 96.8 100 10.32007 PSID 0.72 19.1 40.5 56.3 95.2 101 9.02009 SCF 0.79 29.8 53.2 65.5 98.2 101 14.82009 PSID 0.73 15.4 39.6 55.9 96.6 102 14.3

Notes: Table 1 shows the wealth Gini coefficient, the share of wealthheld by the top 1, 5, 10, 50 and 90 wealthiest households, the share ofhouseholds with zero or negative asset holdings in the U.S. economy. Forthe PSID, drop three samples with wealth less than negative 99 milliondollars.

Table 2 summarizes annualized growth rates of the average level of net worth, earn-

ings, income, and consumption between 2005 and 2007 over 2005 wealth quintiles. It also

documents percentage point differences in expenditure rates which are defined as total

spending as a fraction of disposable income, over the sample period.8 Earnings are the sum

of after-tax wages and salaries, bonuses, overtime, tips, commissions, and transfers. Dis-

posable income is defined as the sum of labor income, unemployment benefits and income

from assets minus federal and state income taxes calculated using the NBER TAXSIM

calculator.9

6See Appendix A for details on sample selection and wealth categories.7Items on expenditure surveyed in the PSID are listed in Appendix A. I also compare the PSID

constructed expenditure data to BEA data in Appendix A.8To study impact of the Great Recession for households with different levels of wealth, I keep households

in each quintile fixed and calculate the annualized percentage change of the averages between two yearsfollowing Krueger et al. (2016).

9Note that, in the PSID, income and earnings data are retrospective while reporting unit for consump-tion varies by items.

8

Table 2. Growth rates of variables across wealth quintiles before the Great Recession2005-2007

NW Quantile Net worth Earnings Income Expenditure Expen. rateQ1(poor) n/a 3.3 3.3 11 3.6Q2 25 1.7 1.9 7.1 2.3Q3 22 1.8 2.0 3.0 0.5Q4 11 1.3 2.9 2.9 0.0Q5(wealthy) 2.1 2.6 1.2 4.3 1.5all 6.8 2.2 2.2 5.1 1.4

Notes : Table 2 shows annualized growth rates of the average net worth, earnings, anddisposable income and percentage points change in expenditure rates from the PSID.

Towards understanding the growth rates of wealth, income and consumption prior to

the Great Recession, Table 2 reports annualized growth rates in these series between 2005

and 2007. The last row shows that aggregate net worth increases by 7 percent on average

per year between 2005 and 2007 while aggregate earnings and income rise by 2.2 percent.

Aggregate consumption rises by more than 5 percent annually and the average consumption

rate rises by 1.4 percent over the sample period. Table 2 also reveals a few facts regarding

the dynamics of each variable over wealth. First, it shows declining growth rates of net

worth across wealth quintiles. This may be driven by mean-reversion which implies a higher

probability of the rise in earnings or income for those who are currently poor compared to

thr rich. Upward bias in growth rates associated with initial small levels of wealth may

also contribute. Second, my PSID constructed spending data shows that growth rates of

consumption generally decline in wealth (column 5). For instance, consumption grows the

most for the bottom 20 percent of households and the least for households in the fourth

quintile of the wealth distribution.10 Finally, the poorest 20 percent of households in 2005

experienced the largest increase in their spending rates and most households increased their

spending rates between 2005 and 2007.

Table 3 summarizes annualized growth rates of net worth, earnings, income, and con-

sumption between 2007 and 2011 to study how wealthy and poor households were affected

during the Great Recession. It reports the growth of the average of each category for the

2007 wealth quintiles. As seen in the last row of Table 3, in the aggregate, net worth fell

10Krueger et al. (2016) also find consumption growth rates falling in wealth over the same sample period.

9

approximately by 2 percent while earnings and income exhibited little growth. Next, in

the aggregate, consumption changes little and the expenditure rate fell by 0.1 percentage

point.11 Thus, overall growth rates of wealth, income and consumption fell over this period

compared to the two years preceding.

Table 3. Growth rates of variables across wealth quintiles during the Great Recession2007-2011

NW Quantile Net worth Earnings Income Expenditure Expen. rate(pp)Q1(poor) n/a 2.1 2.5 0.2 -2.3Q2 11 1.1 0.8 1.9 1.0Q3 -5.1 -0.3 -0.1 -0.2 -0.1Q4 -0.5 -0.7 -0.1 -1.6 -1.7Q5(wealthy) -3.5 0.0 0.3 -2.5 -2.8all -1.7 0.0 0.1 -0.1 -0.1

Notes : Table 3 shows annualized growth rates of the average net worth, earnings, and disposableincome and percentage points change in expenditure rates from the PSID.

Across households of different wealth levels, Table 3 shows net worth growth rates de-

creasing in wealth and becoming negative for the richest 60 percent of households. House-

holds in the third and fifth quintiles experience approximately 5 and 3.5 percent declines in

their wealth while households in the second quintile saw more than 10 percent rise in their

wealth. I also observe rich households having lower growth rates of earnings and income

relative to wealth poor households (third and fourth columns of Table 3). Consumption

also falls more for rich households during the Great Recession, declining by 2.5 percent for

the top 20 percent of households. Lastly, I find a fall in expenditure rates for most house-

holds but the second quintile, which implies that most households increase their savings

rates over the recession.12

To study the severity of the Great Recession across the wealth distribution, Table 4

presents differences in the growth rates of variables between normal times (2005-2007)

and recession (2007-2011). For net worth, households with less wealth tend to experience

11In comparison, Krueger et al. (2016) report a 1.6 percentage points drop in the expenditure rate.12Over 2007-2011, Krueger et al. (2016) also find net worth growth rates falling in wealth as well as

negative consumption growth rates for the top distribution of households. Moreover, they also report risein saving rates across all wealth quintiles.

10

a larger drop in their growth rates of wealth. For example, the poorest 20 percent of

households experience 14 percent decline in their wealth while the richest 20 percent of

households experience 6 percent of decline. Turning to earnings and income, while Table

3 shows a decline only for the third and fourth quintiles over the recession, earnings and

income growth slow down for all households, with the largest fall for the top 20 percent.

Consumption growth rates decline across all wealth quintiles and, most importantly, all

households decrease their expenditure rates with the largest fall for the bottom 20 percent

of households. 13

Table 4. Changes in growth rates of variables between prior- and during the GreatRecession

NW Quantile Net worth Earnings Income Expenditure Expen. rateQ1(poor) n/a -1.2 -0.8 -11 -5.9Q2 -14 -0.6 -1.1 -5.2 -1.3Q3 -27 -2.1 -2.1 -3.2 -0.6Q4 -12 -2.0 -3.0 -4.5 -1.7Q5(wealthy) -6 -2.6 -0.9 -6.8 -4.3all -5.1 -2.2 -2.1 -5.2 -1.5

Notes : Table 4 shows changes in annualized growth rates of the average net worth, earnings,disposable income, and percentage points change in expenditure rates from the PSID.

The fall in expenditure rates across wealth quintiles during the Great Recession com-

pared to normal times is hard to explain in a standard model. Consumption smoothing

motives lead households to raise their spending rates in a recession and the fall in consump-

tion tends to be less than that in income. However, PSID data shows a rise in savings rates

for most households, which likely reflects a strong precautionary savings motive. Below, I

introduce disaster risk in my model and calibrate it to explain this change in saving rates.

13Likewise, Krueger et al. (2016) find the largest drop in expenditure rate for households in the bottomquintile.

11

4 Model

In my model economy, there are four agents: households, an investment firm, a produc-

tion firm, and a government. Households differ by age, productivity, employment status,

and wealth. Household wealth is itself the sum of low-yield liquid assets and high-yield

illiquid assets. Each period, a household chooses its consumption, total saving, and asset

portfolio. It has to pay an idiosyncratic transaction cost if it chooses to cash in its illiquid

wealth and re-balance its portfolio. Illiquid wealth is capital and is held in an investment

firm, which rents it to the production firm. Government supplies the net quantity of the

liquid asset. Interest payments on which are financed by tax revenue.

In this section, I first describe households’ decision problems. Second, I discuss the pro-

duction economy and government. Lastly, I present the definition of recursive equilibrium.

4.1 Households

Households live for a finite number of periods. They enter the labor market at age

j = 1, retire at age Jr = 35, and their last possible age is J = 60. While working, house-

holds face a stochastic idiosyncratic unemployment risk e(z) ∈ {0, pe(z), 1} where z is the

exogenous aggregate state which consists of a total factor productivity shock η and disaster

state d. This unemployment risk, which moves as a function of the exogenous aggregate

state, determines households’ working time in each period.14 Households can be partially

(e = pe) or completely (e = 0) unemployed with probability πu(z) while they are full-time

workers (e = 1) with probability 1 − πu(z). A fraction πp(z) of those not-fully-employed

are working for a fraction pe(z) of a period and unemployed for the remaining time. In

contrast, not fully employed households are laid off for the full period with probability

1 − πp(z). Summarizing, the probability for each idiosyncratic employment status is as

follows:

a full-time worker : 1− πu(z)

14Following Khan et al. (2016), partial unemployment is introduced to match the mean and medianduration of unemployment less than a model period of one year.

12

a part-time worker : πu(z)πp(z)

unemployed : πu(z)(1− πp(z))

Partially and fully unemployed workers receive unemployment benefits from the govern-

ment proportional to their possible earnings and unemployment duration in the period;

the replacement rate is θu. Households draw idiosyncratic productivity shocks ε each pe-

riod, which follow a Markov chain ε ∈ {ε1, ..., εnε}, where Pr(ε′ = εk|ε = εl) = πlk ≥ 0

and∑nε

k=1 πlk = 1.15 These idiosyncratic productivity shocks alongside a labor market

experience premium, l(j), determine the total efficiency units of labor for workers. After

retirement, households receive social security benefits proportional to their last earnings

shock s(ϵJr−1).

As mentioned already, the economy has two assets - a high-yield illiquid asset, a, and

a low-yield liquid asset, b. Each unit of illiquid wealth pays a dividend d(z) and has an

ex-dividend price p(z). Adjusting illiquid assets to a value other than a non-adjusted post-

dividend balance involves an idiosyncratic fixed adjustment cost, ξ, denominated in units

of output, and drawn from a time-invariant distribution, H(ξ). Households paying these

costs adjust illiquid wealth to a desired value; otherwise, the after-tax dividend payments

are re-invested in illiquid wealth.

Borrowing is only allowed in the liquid asset, which is supplied by the government

at a price q. Households face borrowing limits that are a fixed fraction of their age-

varying natural debt limits, ϕbj.16 Note that ϕ is common across households of different

age. Following Kim (2016), I derive age-varying natural debt limits bj in an overlapping-

generations economy.17 Given a natural debt limit for the next age bj+1 and a lowest

possible future earnings xj+1, a natural debt limit for age j is defined as:

bj ≥ qbj+1 − xj+1

15I combined persistent and transitory productivity shocks into one shock process for ease of notation.16In a life-cycle model, the natural debt limit falls with age as households borrow against their future

income.17Given that borrowing is not allowed at the last possible age, I can solve natural debt limits backward

by age.

13

This implies that the maximum debt a household can borrow is the discounted value of

the maximum debt it can borrow tomorrow and the lowest possible future earnings. Age-

specific natural debt limits are crucial for explaining more indebted younger generations as

they allow households to borrow against future income. In the following, ϕ < 1

Each period, a household, identified by its age j ∈ J = {1, ..., J}, illiquid asset, a ∈ A ⊂

R+, liquid asset, b ∈ B ⊂ R, productivity, ε ∈ E, and working status e(z) ∈ {0, pe(z), 1},

chooses consumption and savings. Savings is in two assets, and adjusting the stock of

illiquid wealth requires payment of a transactions cost. A household adjusts its illiquid asset

from the otherwise non-adjusted post-dividend balance if it pays the current idiosyncratic

cost, ξ ∈ Ξ. If a household does not pay its fixed cost, it can only choose consumption and

liquid savings in the current period. The distribution of households, µ, is over (j, a, b, e, ε)

and evolves following the mapping µ′ = Γ(z, µ) where z = (η, d) is a vector summarizing

the exogenous aggregate state.

I now present the problem solved by households. For the ease of notation, I define

transition probabilities for aggregate state vector z by Pr(z′ = (η, d)g|z = (η, d)f ) =

πzfg ≥ 0. Each period, a working household has three idiosyncratic shocks; unemployment

shock e, productivity shock ε, and portfolio adjustment cost ξ. If a household pays its

current fixed cost ξ, it actively adjusts its illiquid wealth account a′. After any adjustment

decision, a household at age j with illiquid assets a, liquid assets b, unemployment shock e,

productivity shock εl, and adjustment cost ξ realizes the value vj(a, b, e, εl, ξ; zf , µ) given

the aggregate state (zf , µ).

vj(a, b, e, εl, ξ; zf , µ) = max

{vaj (a, b, e, εl, ξ; zf , µ), v

nj (a, b, e, εl; zf , µ)

}(1)

where vaj represents the value of a household adjusting its risky illiquid assets and vnj is the

value of a non-adjusting household. Let vej (a, b, e, εl; z, µ) =∫ ξ

0vj(a, b, e, εl, ξ; z, µ)H(dξ) be

the expected value of a household before the idiosyncratic adjustment cost ξ is realized.

A worker receives labor income and an unemployed worker receives unemployment ben-

efits. A retiree receives a proportional social security benefits, s(ϵJr−1). An actively ad-

justing household, paying a resource adjustment cost, cashes in the total stock of wealth

14

in both assets and re-balance its portfolio. Assuming Epstein-Zin preferences, I describe

the optimization problem of a household who pays its adjustment cost to actively adjust

its asset portfolios between high-yield illiquid and low-yield liquid assets.

vaj (a, b, e, εl, ξ; zf , µ) =

maxc, a′, b′

(1− β)c1−σ + β

{nε∑k=1

πlk

nz∑g=1

πzfg

1∑e=0

πe(zg)vej+1(a

′, b′, e′, εk; zg, µ′)1−γ

} 1−σ1−γ

1

1−σ

subject to

c+ q(zf , µ)b′ + p(zf , µ)a

′ ≤ b+ (p(zf , µ) + (1− τa)d(zf , µ))a+ x− ξ (2)

x =

(1− τn)w(zf , µ)l(j)εl(e+ (1− e)θu) if j < Jr

(1− τn)s(ϵJr−1) otherwise

b′ ≥ ϕbj, a′ ≥ 0, c ≥ 0

µ′ = Γ(zf , µ)

where bj is the age-varying natural debt limit.

If a household does not pay its fixed cost, it can only choose its stock of safe liquid

wealth for next period, b′. Illiquid assets pay after-tax dividends which are re-invested.

vnj (a, b, e, εl; zf , µ) =

maxc, b′

(1− β)c1−σ + β

{nε∑k=1

πlk

nz∑g=1

πfg

1∑e=0

πe(zg)vej+1(a

′, b′, e′, εk; zg, µ′)1−γ

} 1−σ1−γ

1

1−σ

subject to

c+ q(zf , µ)b′ ≤ b+ x (3)

15

x =

(1− τn)w(zf , µ)l(j)εl(e+ (1− e)θu) if j < Jr

(1− τn)s(ϵJr−1) otherwise

a′ = (1 + (1− τa))d(zf , µ)a

b′ ≥ ϕbj, c ≥ 0

µ′ = Γ(zf , µ)

4.2 Production and government

The exogenous aggregate state z is summarized by a total factor productivity shock

η and disaster state d. Exogenous total factor productivity, η, follows a Markov chain

η ∈ {η1, ..., ηnη} with Pr(η′|η) = πη,η′ ≥ 0. The economy can be either in a normal

(d = 1), high risk of disaster (d = 2), or disaster state (d = 3). A high risk of disaster

involves a higher probability of the disaster state next period. The disaster state has

an additional drop in TFP, determined by the parameter λ, conditional on the economy

already experiencing a fall in TFP. 18 TFP is (1 − λ(d, η))η, where λ(3, η = η1) = λ < 1

and λ(3, η) = 0 otherwise. I assume that d also follows a Markov chain d ∈ {1, 2, 3} with

πd =

p11 p12 p13

p21 p22 p23

p31 p32 p33

where pij is the probability of transiting from i state to j state.

A competitive investment firm sells shares of total capital at an ex-dividend price p(z, µ)

and pays dividends d(z, µ) to households where µ is the distribution of households. The

investment firm owns the technology that creates capital and rents this capital to a pro-

duction firm at a rental rate rk(z, µ). Specifically, investment firm sells shares to future

capital, p(z, µ)k′, to households to transform k′ units of current output into capital for the

18I will assume that the disaster state affects TFP when the latter falls by more than one standarddeviation from its mean. This will be the case only when η = η1.

16

next period. Moreover, the investment firm faces convex capital adjustment costs.

Φ(k′, k) =

(k′ − k

k

)2

k

This adjustment cost introduces deviation in the price of capital from that of consump-

tion as it makes consumption and capital less than perfectly substitutable. I describe the

optimization problem of the investment firm.

J(k, zf , µ) = maxk′

((rk(zf , µ) + 1− δ)k − (p(zf , µ) + d(zf , µ))k

+ p(zf , µ)k′ − k′ − Φ(k′, k) +

nz∑g=1

πfgr(zg, zf , µ)J(k′, zg, µ

′)

)

where investment firm discounts future earnings by r(zg, zf , µ).19

The production firm employs capital k and hires labor n to produce output through a

CRS production function y = (1− λ(d, η))ηkαn1−α, where 0 < α < 1 and η > 0. Thus, the

optimization problem of the production firm is

maxk,n

((1− λ(d))ηkαn1−α − rk(z, µ)k − w(z, µ)n

)The government supplies a quantity of liquid assets, B, at a price q(z, µ). Labor income

is taxed at τn and dividend income at τa. Social security benefits and unemployment

benefits are also taxed at τn. Government revenues are used to finance social security benefit

payments, unemployment benefits payments, interest payments on debt, and government

spending, G(z, µ) ≥ 0.20

19In equilibrium, this discount factor should be equal to the marginal rate of substitution of householdswho are holding illiquid assets. In my model economy, it is hard to solve as households endogenously holddifferent asset portfolios. In Appendix B, I will show how the equilibrium prices and dividend paymentsto the shares of capital are not affected by this discount factor.

20Every period, spending is chosen to balance the government budget.

17

4.3 Recursive equilibrium

Define the product space S = J×A×B×E×{0, pe, 1} for the distribution of households.

Given the Borel algebra S generated by the open subsets of S, µ : S → [0, 1] is a probability

measure over households. Households start with an initial wealth of zero and an initial labor

productivity drawn from π0 ∼ logN(0, σ2π).

A recursive competitive equilibrium is a set of functions

(v, va, vn, ca, cn, ha, hn, ba, bn, χ, k, n, r, q, w)

such that:

(i) (v, va, vn) solves (1)−(3), and (ca, ha, ba) are the policy functions associated with (2) for

consumption, illiquid and liquid asset savings by a household that adjusts its illiquid

asset holdings. (cn, hn, bn) are the policy functions associated with (3) for consumption

and savings in illiquid and liquid assets by a non-adjusting household. χ is the decision

rule associated with (1), and χ = 1 when the fixed cost to adjust illiquid assets is paid.

(ii) The government budget is balanced

G(z, µ)+Bs+1∑

e=0

J∑j=1

nε∑l=1

∫A

∫B

(1−τn)(s(εl1j≥Jr)+(1−e)θuwl(j)ε1j<Jr)µ(j, da, db, e, εl)

= τad(z, µ)k + τnw(z, µ)n+ q(z, µ)B′

s

(iii) Markets clear

n(z, µ) =1∑

e=0

J∑j=1

nε∑l=1

∫A

∫B

l(j)εleµ(j, da, db, e, εl)

k(z, µ) =1∑

e=0

J∑j=1

nε∑l=1

∫A

∫B

aµ(j, da, db, e, εl)

B(z, µ) =1∑

e=0

J∑j=1

nε∑l=1

∫A

∫B

bµ(j, da, db, e, εl)

18

(iv) Prices are competitively determined

w(z, µ) = (1− α)(1− λ(d))ηkαn−α

rk(z, µ) = α(1− λ(d))ηkα−1n1−α

p(z, µ) = 1 + Φ1(Gk(z, µ), k)

d(z, µ) = α(1− λ(d))ηkα−1n1−α − δ − Φ1(Gk(z, µ), k)− Φ2(Gk(z, µ), k)

where Gk(z, µ) is the aggregate law of motion for aggregate capital. Φ1 and Φ2 are the

derivatives of Φ with respect to Gk and k, respectively.2122

(v)

µ′(j + 1, A0, B0, e′, εk) =

πe′(z)nε∑l=1

πlk

(∫∆1

µ(j, da, db, e, εl)H(dξ) +

∫∆2

µ(j, da, db, e, εl)H(dξ)

)∀ j

where ∆1 = {(a, b, e, εl, ξ)|ha(j, a, b, e, εl, ξ; z, µ) ∈ A0, ba(j, a, b, e, εl, ξ; z, µ) ∈ B0 and χ(j, a, b, e, εl, ξ; z, µ) = 1}

and ∆2 = {(a, b, e, εl, ξ)|hn(j, a, b, e, εl; z, µ) ∈ A0, bn(j, a, b, e, εl; z, µ) ∈ B0 and χ(j, a, b, e, εl, ξ; z, µ) = 0},

(j, a, b, e, εl) ∈ S and ξ ∈ Ξ.

5 Calibration

In order to bring the model to data, I calibrate the model economy using household

data. Given its detailed wealth information, I use the 2007-2009 SCF panel data to cali-

brate parameters directly affecting asset holdings including the distribution of adjustment

cost. Parameters governing unemployment are calibrated to match the mean and median

unemployment duration in the CPS and the average unemployment rate in the BLS. Lastly,

21Note that these equilibrium price function, using Φ(k′, k) =(

k′−kk

)2

k imply p = 1 and d =

αηkα−1n1−α − δ in steady state.22Given the definition of recursive competitive equilibrium, I show the prices, p(z, µ) and d(z, µ) are

consistent with equilibrium in Appendix C.

19

I estimate the earnings shock process using the PSID.

In the SCF, I define illiquid wealth as stocks, business equity, net residential property,

net equity in non-residential real estate and net consumer durables.23 The remaining assets

and debt are considered liquid following Kaplan et al. (2016). I calibrate β to match the

capital (productive illiquid asset) to output ratio of 2.66. Following Kaplan et al. (2016), I

consider business equity, stocks and net equity in non-residential real estate as productive

illiquid assets as well as 40 percent of net housing and consumer durables.24 Given that

a sampling unit in the SCF is a household, the total value of productive illiquid assets is

divided by the average family size in 2007 to make it comparable to GDP per capita.25 The

calibrated economy implies a 6.4 percent return on illiquid wealth in the steady state when

a zero return on liquid wealth is targeted, resulting in a liquidity premium of 6.4 percent.

The capital share of output is α = 0.36. Despite the share of liquid wealth to GDP not

being a target, the calibrated economy explains a ratio of 29 percent, which is close to the

35 percent in the 2007 SCF.

I assume a model period of one-year. Households enter the labor market at age 25,

retire at age 60, and live until age 84 with certainty. I assume Epstein-Zin preferences,

allowing the elasticity of intertemporal substitution (EIS) to deviate from the inverse of

the coefficient of relative risk aversion. I set the coefficient of relative risk aversion to 2 and

the EIS is 1.5.26 Social security payments are paid based on the average of the highest 35

years of earning by Social Security Administration (SSA). In the model, calculating average

23Following Glover et al. (2016), money market mutual funds and quasi-liquid retirement accounts areincluded in stocks. In the absence of a housing market and collateralized borrowing, I include residentialproperty net of all debt secured by residential property (mortgages, home equity loans, and HELOCs)following Kaplan and Violante (2014). By defining housing wealth as a net value, I abstract from the issueof home equity loans which provide liquidity through housing asset.

24Kaplan et al. (2016) argue that a fraction of housing and consumer durables could be rented to businessor used in production. I follow their approach.

25As GDP per capita is expressed in chain 2009 dollars, I adjust the value of productive illiquid wealthin 2009 dollars using CPI-IPUMS.

26Gourio (2012) shows that it is important to have the EIS greater than 1 in a model with a disasterrisk to reproduce a countercyclical risk premia as seen in the data. He also shows that, in an economy withstandard expected utility, investment is much less volatile than the data. As noted by Kaplan and Violante(2014), when regressing consumption growth on real interest rates, an estimate of the EIS is downward-biased because of measurement error and endogeneity issues. By contrast, the estimates derived usingGMM are generally greater than 1. Similarly, Van Binsbergen et al. (2012) estimated a value of the EISabove 1 in a DSGE production economy in which households have Epstein-Zin preferences using maximumlikelihood.

20

earnings requires one more state variable, making computation more challenging. Given the

high persistent earnings process, I proxy the history of earnings over a worker’s life-cycle us-

ing the level of earnings in the last working period, s(ϵJr−1) = θsw∑Jr−1

j=1 l(j)

Jr−1εJr−1. Following

Hosseini (2015), θs is chosen to match the replacement rate of 45 percent of average pre-tax

earnings in the steady state. Labor income is taxed at 27 percent (Domeij and Heathcote

(2004)). I chose τa to imply a 25 percent capital income tax rate in a steady state.27

I select parameters p, πu, πp to match mean and median unemployment durations as

well as the unemployment rate as in Khan et al. (2016). The working period for a partially

employed household p is chosen to match the median unemployment duration of 12 weeks

between 1981 and 2016 in the CPS.28 The probability of a partial employment πp varies

with a TFP shock, πp ∈ [πp−εp, πp+εp]. A steady state level of the probability of a partial

employment πp is chosen to match the mean unemployment duration of 24 weeks between

1981 and 2016.29 The mean duration of unemployment rises to 36 weeks after 2008 which

is around 60 percent of a model period. I select εp to match this rise in duration when

the economy is in recession.30 The probability of unemployment πu also varies with a TFP

shock, πu ∈ [πu − εlu, πu + εhu]. πu is chosen to match an unemployment rate of 5 percent in

the steady state. εlu is chosen to explain a 5 percent rise in the unemployment rate over the

Great Recession while εhu results in a 3 percent unemployment rate when the economy is in

a boom.31 Lastly, I choose 43.5 percent for a replacement rate of unemployment benefits

θu following Nakajima (2012).

The idiosyncratic earnings shock ε consists of both a persistent and a transitory com-

ponent. The persistent shock follows an AR(1) process;

ε = φ+ εv,

27Landvoigt et al. (2016) estimated a capital income tax rate of 20 percent from government corporatetax revenue as a share of GDP using BEA data. In Jermann and Quadrini (2012), the average tax rate is35 percent. I chose a value in this range.

28 Partial unemployment is introduced to handle the median unemployment duration less than a modelperiod. As one year has 52 weeks, the median unemployment duration corresponds to 23 percent of amodel period.

29Note that the mean unemployment duration is calculated as 2452 = πp(1− p) + (1− πp) ∗ 1.

30Given p, this implies that 3652 = (πp − εp)(1− p) + (1− (πp − εp)) ∗ 1

31Khan and Thomas (2013) measured a TFP drop by 2.18 percent in the recent recession. Thus, εu ischosen to imply 10 percent of unemployment rate when a TFP falls by 2.18 percent.

21

φ′ = ρεφ+ εs,

where εs ∼ N(0, σ2s) and εv ∼ N(0, σ2

v). The values of ρε, σ2s , σ

2v are the 2007 estimates

in Khan and Kim (2015). They also estimated a labor market experience premium l(j) by

running an OLS regression of the log of hourly wages on time dummies; an interaction term

with education and time dummies; labor market experience approximated by age minus

years of schooling minus 5; and experience-squared. Figure 1 illustrates the resulting labor

market experience premium.

Figure 1. Labor market experience premium

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58

l(j)

Number of years in labor market

The distribution of idiosyncratic fixed costs ζ ∈ [0, ζ] is assumed to be drawn from

an uniform distribution. Households can borrow up to a fixed fraction ϕ of their age-

varying natural debt limits, bj. I choose these parameters (ζ, ϕ) to match the share of

households holding zero or negative net worth and the share of households holding positive

illiquid assets but little liquid assets as seen in the 2007 SCF.These liquidity constrained

households are comparable to poor and wealthy hand-to-mouth households respectively as

22

defined in Kaplan and Violante (2014) and Kaplan et al. (2016).32 Kaplan and Violante

(2014) point out that these households have a high response in consumption following a

transitory income change.33 Thus, these parsimonious targets have the virtue of allowing

for the mechanism that Kaplan and Violante (2014) emphasize: a potentially important

role for liquidity constrained households in driving a large aggregate consumption response.

Borrowing limits are important in determining the number of liquidity constrained

households as these households have little liquid assets. Figure 2 shows the resulting age-

specific borrowing limits which, in contrast to the standard age-invariant limit, allows

more borrowing by younger cohorts. Thus, I calibrate parameters (ζ, ϕ) to match liquidity

constrained households in the 2007 SCF. As seen in the next section, given this calibration

strategy, the calibrated economy explains a significant fraction of the distributions of the

net worth as well as illiquid and liquid wealth. Furthermore, the benchmark economy gives

rise to the share of illiquid assets to net worth over wealth deciles and age cohorts without

being a calibration target.

Aggregate TFP is assumed to follow an AR(1) process in logs, log η′= ρη log η+εη where

εη ∼ N(0, σ2η). Khan and Thomas (2013) estimated this process using Solow residuals

calculated from data on real U.S. GDP, private capital, and total employment hours. I use

their estimates and discretize this process into nη = 3.34

For the transition probabilities for disaster risk, I first assume that the probability of

staying in normal times is equal to the persistence of ordinary TFP shocks. Next, in order

to make disaster a rare event, I set the probability of going to the disaster state from

a normal state to a negligible value, p13 = 0. Next, I assume that the disaster state is

32Kaplan and Violante (2014) defined wealthy hand-to-mouth (htm) households as those with liquidwealth less than half of their earnings but holding positive balances of illiquid wealth. They estimated atotal fraction of hand-to-mouth households between 17.5 percent and 35 percent of households in the 2004SCF, including both wealthy and poor households. Among htm households, they find 40 to 80 percentare wealthy. I instead define wealthy htm households as those with no liquid wealth but positive illiquidwealth.

33Kaplan and Violante (2014) showed that the average MPC for HTM households is around 40 percentwhile that for non-HTM households is only 7 percent.

34Khan and Thomas (2013) estimated an AR(1) TFP shock process in a an yearly model with the samecapital depreciation rate and the capital share in production function as my model economy. Though theyassumed a DRS technology instead of a CRS technology, this difference only slightly affects the standarddeviation of a TFP shock process.

23

preceded by a rise in the risk of disaster, which itself has a higher probability of transiting

into a disaster state. Imposing the following symmetry conditions, p11 = p33, p12 = p32,

p13 = p31, p21 = p23, I only need to set one more parameter: the probability of staying in a

high-risk state p22. Given that it is hard to pin down the transition probabilities in a high

disaster risk state, I assume the following transition probability matrix:

πd =

0.909 0.091 0

0.25 0.5 0.25

0 0.091 0.909

Disaster risk is usually defined as wars or economic depressions which result in more than

a 15 percent decline in real GDP per capita (Barro, 2006) and modelled as a shock to TFP

or capital destruction. (Gourio, 2012). λ is chosen to match a drop in the expenditure rate

for the bottom 20 percent of households in the Great Recession.35

5.1 Numerical Overview

I develop a two-stage approach to solve savings decisions with two assets. In the first

stage, given a current fixed cost, a household chooses whether or not to adjust its portfolio.

If it adjusts, it chooses its savings in illiquid wealth, a′. In the second stage, given a′, I solve

for the optimal choice of liquid wealth, b′, using the endogenous grid method (Carroll, 2006).

To solve the model with aggregate uncertainty, I extend the Backward Induction Method

of Reiter (2002, 2010). This involves generalizing the method to solve an OLG economy

with bivariate cross-sectional distribution of continuous endogenous state variables. The

Backward induction method of Reiter allows the distribution of households to vary in

potentially rich ways as a function of an approximate aggregate state as it does not impose

a parametric aggregate law of motion. Solving individual decision rules and the consistent

aggregate law of motion simultaneously, it does not require repeated simulation, reducing

computation time compared to Krusell and Smith (1998). Please see Appendix D for more

details.

35Results are not very sensitive to the transition probabilities parameters for disaster state. Details areavailable in next revision.

24

Figure 2. Borrowing limits

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

25

28

31

34

37

40

43

46

49

52

55

58

61

64

67

70

73

76

79

82

neg

ativ

e li

quid

wea

lth

age

25

Tab

le5:

Summaryof

param

etrization

Param

eterssetexternally

Value

αcapital

shareof

output(NIPA)

0.36

δdep

reciationrate(N

IPA)

0.069

τ n,τ

alabor

andcapital

incometaxes

0.27,0.25

(ρη,σ

η)

TFP

shock

process

(Khan

andThom

as,20

13)

(0.909,0.014)

σcoeffi

cientof

relativerisk

aversion

2.0

γinverseof

EIS

1 1.5

λ(d

=3)

additional

dropin

tfpain

disasterstate

0.1

θ sreplacementrate

ofav

gpre-tax

earnings

forsocial

secu

rity

0.45

θ ureplacementrate

ofav

gpre-tax

earnings

forunem

ploymentben

efit

0.435

(σs,σ

v,ρ

ε)

earnings

shock

process

(Khan

andKim

,20

16)

(0.2449,0.3937,0.9825)

l(j)

malehou

rlywag

elife-cycleprofile

seetext

Parameterscalibrated

Value

mom

ents

tomatch

data

model

β0.937

capital

toou

tputratio

2.66

2.69

pe

0.22

87med

ianunem

ploymentduration

asafraction

ofamodel

period

0.22

870.2287

πd

seetext

tran

sition

probab

ilitymatrixfordisasterstate

πe(z)

seetext

unem

ploymentrate

ζ1.4

shareof

hhsholdingpositiveilliquid

assetbutlittle

liquid

asset

0.32

0.32

ϕ0.4

shareof

hhsholdingzero

ornegativenet

worth

0.10

30.103

26

6 Results

I begin by discussing the steady state of the model economy. Next, I discuss the

implications of wealth composition heterogeneity for business cycles driven by shocks to

TFP and changes in disaster risk. Lastly, I evaluate predictions for changes in the joint

household distribution of net worth, earnings, income and consumption.

6.1 Steady state

Table 6 summarizes distributions of net worth, illiquid asset, and liquid assets in the

2007 SCF and the steady state of the economy with heterogeneity in both the level and com-

position of net worth. Though it only targets the share of liquidity constrained households,

the portfolio choice economy successfully reproduces much of the dispersion in wealth. As

seen in the top panel, the quintile distributions of net worth in the benchmark (wealth

composition heterogeneity) economy are broadly consistent with those in the 2007 SCF,

explaining more than 70 percent of the total wealth held by the wealthiest 20 percent of

households. The wealth Gini is within 9 percentage points of its empirical counterpart.36 As

pointed out in Krueger et al. (2016), reproducing enough dispersion in wealth in a model is

important for explaining aggregate dynamics.37 A realistic skewness in the distribution of

wealth is also crucial for studying changes in the cross-sectional distributions of consump-

tion and income as the recent recession had disparate effects on households with different

levels of total wealth as shown in section 3.38

What explains differences between rich and poor households in the steady state of the

benchmark economy? First, households with a favorable earnings shock have a propensity

36This is not driven by a high fraction of households near the borrowing limit as in Huggett (1996) butby the right tail of the distribution. The wealthiest 10 percent of households hold 57 percent of wealthwhile they hold 64 percent in the 2007 SCF.

37Krueger et al. (2016) show that the drop in aggregate consumption is 0.5 percentage points larger inan economy with more wealth inequality than in a representative agent economy.

38While interesting, explaining the very right tail of the distribution is not of primary importance asthe analysis largely focuses on the quintile distributions of wealth where the model matches the data fairlywell.

27

Table 6. Distributions of net worth, illiquid and liquid assets

Net worth Q1 Q2 Q3 Q4 Q5 top 1% 5% 10% ≤ 0 Gini2007 SCF -0.3 1.4 5.7 14.1 79.1 29.1 52.3 64.3 10.3 0.78Benchmark 0.2 3.8 8.9 16.1 71.1 11.6 38.2 57.1 10.3 0.69Single asset 1.1 4.9 9.7 16.7 67.6 9.9 34.5 52.3 2.16 0.64

Illiquid wealth Q1 Q2 Q3 Q4 Q5 top 1% 5% 10% Gini2007 SCF 0.14 1.6 5.9 14.4 78.0 28.2 51.2 63.2 0.76Benchmark 0.0 3.0 8.1 15.3 72.0 11.9 40.3 59.1 0.71

Liquid wealth Q1 Q2 Q3 Q4 Q5 top 1% 5% 10% Gini2007 SCF -11.7 -0.53 0.92 7.8 103 47.1 76.2 90.5 0.92Benchmark -8.3 -1.4 4.89 22.9 82.0 11.3 36.7 54.9 0.90

Notes : Table 6 shows the share of net worth, illiquid asset, and liquid asset across the wealthquintiles. It also reports the share held by the top 1, 5, and 10 households for net worth, illiquidasset, and liquid asset. Lastly, it reports the share of households with zero or negative net worth inthe 2007 SCF and model and the Gini coefficient for net worth, illiquid asset, and liquid asset.

to smooth consumption by accumulating a higher level of wealth compared to others.

Second, a household with a high level of earnings can afford portfolio adjustment costs and

invest in illiquid assets to receive a higher return on savings which accelerates its wealth

accumulation. Furthermore, fixed transaction costs realized to adjust illiquid asset holdings

discourage households from investing in illiquid assets until they are sufficiently wealthy.

This endogeneity in the composition of wealth is crucial for explaining asset positions

before the recession and the subsequent heterogeneous responses during the recession as

households with different portfolios are dissimilarly affected by changes in asset prices.

Table 6 also summarizes the distributions of high-yield illiquid assets and low-yield liquid

assets in the 2007 SCF and in the portfolio choice economy. The distributions of illiquid

assets are comparable to those in the data, reproducing a Gini of 0.71 and 72 percent of the

illiquid assets held by the top 20 percent of households compared to 78 percent in the data.

However, SCF shows more dispersion in liquid assets. The liquid wealth Gini is around 0.91

and the top 20 percent of households hold all of net liquid assets in the economy. Borrowing

by the bottom 20 percent of households represents approximately 12 percent of the total

stock in the 2007 SCF. Though the distribution of low-yield assets falls short of the data

for the top 10 percent of households, the benchmark economy qualitatively matches the

more skewed distribution of liquid relative to illiquid assets, explaining the liquid wealth

28

Gini of 0.90.

Importantly, age-varying borrowing limits that are a common percentage of age-consistent

natural debt limits, alongside fixed costs, give rise to empirically consistent average shares

of illiquid assets across households of different levels of wealth and age. As seen in Figure

3, the benchmark economy predicts more than 80 percent of the total wealth held in illiq-

uid assets for the top 80 percent of households and the high fraction of the wealth held

as illiquid assets for the bottom 11-20 percent of households (2nd decile). The high ratio

of illiquid assets to net worth, for the poorest 11-20 percent of households, is the result

of wealth-poor households borrowing in liquid asset markets. This drives a small level of

wealth which biases up their share of illiquid assets to net worth.

Figure 4 shows the corresponding figure over age groups. Though the benchmark econ-

omy overestimates the average share of total wealth held in illiquid assets by the youngest

cohort, it captures the share’s decline over age. Older cohorts tend to hold a higher fraction

of safe liquid assets compared to younger cohorts as they have less time to recover from

any fall in asset prices.39 In the benchmark economy, age-varying borrowing limits are

critical in explaining the high share of illiquid assets held by the young. More borrowing

by younger households also biases up their share of illiquid assets to net worth embedded

in the small level of wealth.

6.2 Aggregate Dynamics

In this section, I first present dynamic results for the model using TFP shocks. Next, I

introduce disaster risk to show how a rise in the probability of a severe economic recession,

consistent with rational expectations, amplifies the effects of a shock to TFP. Throughout,

I examine the model using two different assumptions about the supply of liquid assets. The

first involves a constant stock of liquid assets (B fixed) while the other assumes a constant

39Glover et al. (2016) also find that younger cohorts hold less safe assets than old cohorts because oftheir large mortgage debt which is considered safe in their classification. I still find the high share of illiquidassets to total wealth by the young when measured using the net value of housing and consumer durables.

29

Figure 3. Share of illiquid asset as a fraction of net worth over wealth deciles

0

0.5

1

1.5

2

2.5

3

3.5

4

2 3 4 5 6 7 8 9 10

wealth deciles

data model

Notes : The average share of illiquid asset as a fraction of net worth over wealth deciles in the

2007 SCF (blue bars) and the benchmark economy (red bars).

return (rf fixed). Thus, in the second case, the supply of liquid assets is perfectly elastic

while, in the first, government policy holds the supply of such assets fixed.40 Note that,

while the return on liquid savings is held constant in equilibrium in a fixed return model,

this is consistent with general equilibrium characterized by the government policy which

adjusts the supply of liquid assets to maintain the constant return.

6.2.1 Heterogeneity in the composition of wealth

In Table 7 and 8, I summarize business cycles moments from simulating two portfolio

choice models.41 In both tables, we see that models with multiple assets of varying liquidity

produce consumption approximately 40 percent as volatile as output and investment twice

as volatile. Furthermore, the portfolio choice economy has a procyclical return on savings

in liquid assets when their stock is fixed (column 9 of Table 7). In contrast, when the

40Each of these assumptions about government policy over the business cycle serves as a benchmark.Over the 2007 recession, real interest rates on safe, liquid assets fell markedly while, at the same time, aflight to quality led to a large rise in their supply. In a future revision, I plan to address this by modellinga countercyclical supply of liquid assets that responds to a higher probability of disaster. The supplyfunction describing government policy can be chosen to ensure a fall in the safe real interest rate consistentwith the data.

41I simulate the model economies with an aggregate productivity shock for 2300 periods and drop thefirst 300 periods.

30

Figure 4. Share of illiquid asset as a fraction of net worth over age groups

0

0.5

1

1.5

2

2.5

25-34 35-44 45-54 55-64 65-74 over 75

age bin

data model

Notes : The average share of illiquid asset as a fraction of net worth over age group in the 2007

SCF (blue bars) and the benchmark economy (red bars).

return is rigid, the equilibrium stock becomes countercyclical (column 6 of Table 8). As a

result, and somewhat surprisingly, the aggregate economy behaves similarly across the two

cases where the supply of liquid assets is perfectly inelastic or elastic.

A rise in unemployment risk, during a recession, increases households’ demand for liquid

assets. Households attempting to smooth consumption following a decline in their earnings

increase their stock of liquid savings relative to illiquid assets. Given a fixed stock of such

assets, portfolio adjustments by households lower their return. When instead the govern-

ment choose to hold the return on liquid savings fixed, this portfolio adjustment drives a

rise in liquid assets over a recession.42

Table 9 shows business cycle moments for the fraction of households who adjust their

illiquid assets and for the share of liquid assets to the total stock of wealth in the economy.

As mentioned above, during a recession, following a reduction in earnings and a rise in un-

42While these forces affect the composition of household savings, their aggregate effects in ordinarybusiness cycles is small. As a result, fluctuations in GDP, consumption, and investment resemble those ina single asset economy subject to the same TFP and unemployment risk shocks. (Table D2)

31

Table 7. Business cycle statistics in a portfolio choice model (B fixed)

x = Y C I K B N E(r) rf wmean(x) 2.48 1.84 0.48 6.45 0.73 1.44 0.07 0.0 1.10σx/σy (2.75) 0.38 1.95 0.22 n/a 0.86 0.27 0.02 0.29

corr(x, y) 1.0 0.95 0.99 -0.04 n/a 0.96 0.94 0.79 0.59

Notes : Table 7 presents means of GDP, consumption, investment in illiquid wealth,stock of capital, supply of liquid wealth, total hours worked, expected return on illiquidwealth, return on liquid savings and wage for the model simulated data. It also listsrelative standard deviation to and correlation with GDP for each variable. I smoothseries using a HP-filter with a smoothing parameter of 100.

Table 8. Business cycle statistics in a portfolio choice model (rf fixed)

x = Y C I K B N E(r) rf wmean(x) 2.49 1.84 0.49 6.50 0.65 1.44 0.07 0.0 1.10σx/σy (2.75) 0.38 2.0 0.22 0.26 0.86 0.56 n/a 0.29

corr(x, y) 1.0 0.96 0.98 -0.02 -0.46 0.96 0.93 n/a 0.59

Notes : Table 8 presents means of GDP, consumption, investment in illiquid wealth,stock of capital, supply of liquid wealth, total hours worked, expected return on illiquidwealth, return on liquid savings and wage for the model simulated data. It also listsrelative standard deviation to and correlation with GDP for each variable. I smoothseries using a HP-filter with a smoothing parameter of 100.

employment, more households increase their holdings of liquid assets to offset the drop in

their consumption, resulting in the counter-cyclical share of liquid assets across households.

In the economy where the supply of such assets is allowed to vary, the share of liquid assets

rises in a recession as their stock increases. This results in a more variable share than when

government policy holds the supply of such assets fixed.

Now, I present impulse response functions for aggregate variables following a drop in

TFP, accompanied by a rise in unemployment risk, in Figures 5 and 6 for a single asset

economy as well as two heterogeneous composition of wealth economies. I assume that TFP

falls for the first two periods then starts to recover at the rate implied by its persistence.43

Following a drop in TFP, output declines by 6 percent on impact both in a model with a

single asset and in models with multiple assets of varying liquidity. Given a predetermined

43I choose 2.18 percent drop in a TFP to be consistent with the estimated decline in measured TFPover the Great Recession reported in Khan and Thomas (2013).

32

Table 9. Business cycle statistics for the share of households adjusting illiquid assets andthe share of liquid assets to total wealth

B fixed rf fixed

x = adjusting pop BK+B

x = adjusting pop BK+B

mean(x) 0.144 0.102 mean(x) 0.147 0.091σx/σy 0.279 0.192 σx/σy 0.272 0.309

corr(x, y) -0.485 0.046 corr(x, y) -0.424 -0.327

Notes : Table 9 presents means of the share of households who adjust their illiquidassets and the share of liquid assets to total wealth for the model simulated data. Italso lists relative standard deviation to and correlation with GDP for each variable.I smooth series using a HP-filter with a smoothing parameter of 100.

stock of capital, the fall in output is driven by the drop in TFP and aggregate labor

supply. As labor supply is exogenous and all economies experience the same 5 percent rise

in unemployment, the change in output on impact must be the same.

In Figure 5 (d), aggregate consumption exhibits a gradual, familiar, hump-shaped re-

sponse. Moreover, aggregate consumption falls more in economies with illiquid assets com-

pared to a single asset economy. For instance, in the absence of any change in the aggregate

stock of liquid assets, aggregate consumption decreases by approximately 2.3 percent while

it only falls by 2 percent in the single asset case. The larger fall in consumption is the result

of illiquidity in wealth. The costs of adjusting illiquid assets deters households’ ability to

smooth consumption. As a result, consumption is more volatile in response to shocks to

TFP and employment.

Adding liquidity risk into a model produces a slow recovery which is one of the char-

acteristics of the Great Recession. In Figure 5 (d), the half-life of consumption in the

economy with a fixed stock of liquid wealth is 1.4 times greater than that in a single asset

model. This slow recovery is the result of the equilibrium fall in the real return to safe

assets which depresses savings and consumption growth. Transactions costs imply a range

of portfolio shares over which households are unwilling to increase their holdings of illiquid

wealth. The slower growth in liquid assets for their non-investors implies fewer households

reaching a sufficiently low share of illiquid wealth to invest in capital.

Aggregate capital and investment fall more in portfolio choice economies following a

33

drop in TFP compared to a single asset economy as seen in Figures 6 (a) and 6 (b). In a

single asset economy, households can raise their buffer stock of savings only through capital,

hastening the recovery. However, in an economy with multiple assets that bear different

risk and liquidity, households, who invested in high-yield assets at the expense of liquidity,

re-balance their portfolios toward liquid assets that offer value in smoothing consumption.

This results in a larger drop in investment and capital compared to a single asset economy.

Table 10 summarizes peak-to-trough declines for each series in the Great Recession and

in model economies. Following a 2.18 percent decline in TFP, an economy with one asset

experiences 10 percent drop in investment and 2.4 percent drop in aggregate consumption

compared to 19 percent and 4 percent, respectively, in data. As seen in the third and fourth

rows of the Table 10, adding another asset that carries liquidity risk explains an additional

2 to 3 percentage points drop in aggregate investment. Although aggregate consumption

falls more by 0.2 percentage points in a fixed liquid asset economy from those in a single

asset economy, portfolio choice economies do not explain enough of the drop in aggregate

consumption seen in data. In the following section, I introduce the disaster risk to study

the amplification mechanism of a rise in risk of a disastrous recession across households.

Table 10. Peak-to-Trough declines: U.S. 2007 Recession and model

GDP I N C TFPdata 5.59 18.98 6.03 4.08 2.18single asset 5.92 10.00 5.58 2.45 2.18two asset (B fixed) 6.06 12.06 5.58 2.61 2.18two asset (rf fixed) 5.98 12.54 5.58 2.42 2.18

Notes : Table 10 shows peak-to-trough declines between 2008q4 and2009q2 in the first row (Khan and Thomas, 2013). The second rowis a single asset economy. The third row is a model with two assets,where the return on liquid wealth is fixed. The last row is a twoasset model with a fixed stock of liquid assets.

34

Figure 5. Impulse response

-2.5

-2

-1.5

-1

-0.5

0

1 11 21 31 41 51 61 71 81 91

per

cen

t ch

ang

e

date

(a) total factor productivity

-6

-5

-4

-3

-2

-1

0

1 11 21 31 41 51 61 71 81 91

per

cen

t ch

ang

e

date

(c) hours worked

-8

-6

-4

-2

0

1 11 21 31 41 51 61 71 81 91

per

cen

t ch

ang

e

date

(b) output

single asset economy

two asset economy (B fixed)

two asset economy (r_f fixed)

-3

-2.5

-2

-1.5

-1

-0.5

0

1 11 21 31 41 51 61 71 81 91

per

cen

t ch

ang

e

date

(d) consumption

Notes : Figure 5 shows aggregate response to a 2.18 percent drop in TFP in a single asset economy

(black solid line), in a heterogeneous asset economy with a fixed quantity of liquid wealth (blue

dashed line), and in a portfolio choice economy with a fixed price of liquid wealth (red dotted

line). The vertical axis measures the percent change of a variable from its mean.

35

Figure 6. Impulse response

Notes : Figure 6 shows aggregate response to a 2.18 percent drop in TFP in a single asset economy

(black solid line), in a heterogeneous asset economy with a fixed quantity of liquid wealth (blue

dashed line), and in a portfolio choice economy with a fixed price of liquid wealth (red dotted

line). The vertical axis measures the percent change of a variable from its mean.

36

6.2.2 Disaster risk

In this section, I introduce aggregate disaster risk into the wealth composition hetero-

geneity economy to explore how a persistent rise in the risk of economic disaster interacts

with difference in liquidity of household wealth. Figure 7 presents impulse responses fol-

lowing a 2.18 percent drop in TFP and a rise in unemployment risk for the first two periods

in a portfolio choice economy where the return on liquid savings is fixed over time while

Figure 8 shows the corresponding figure in a portfolio choice model where the stock of

liquid wealth is fixed. A rise in disaster risk raises households’ desire for liquidity. The

solid line in Figure 7 shows that, when there is no change in disaster risk, the aggregate

response of the economy following a TFP shock is close to those in Figures 5 and 6 where

there was no disaster. This muted effect of aggregate disaster risk in ordinary times is the

result of a rare probability of further deterioration in TFP and earnings.44

In contrast, a rise in the risk of a sharp fall in TFP, across households, implies a different

response in the aggregate economy as seen in Figure 7. First, a rise in disaster risk qualita-

tively changes the familiar hump-shaped response of aggregate consumption in the wealth

composition heterogeneity model with a fixed rate of return on liquid wealth. This is due

to a change in the average substitution effect when the future real rate of return to wealth

is expected to be lower, following a rise in disaster risk, than the current average return.

Over a typical TFP shock driven business cycle, the real return on assets rises over time

following an initial drop. Over this period, the substitution effect leads to further reduc-

tions in consumption even as the negative wealth effect declines, generating a hump-shaped

response in aggregate consumption. In contrast, when the real rate of return on liquid sav-

ings is rigid, the return to portfolios is expected to fall in the future with a rise in disaster

risk. This fall in the expected real return on wealth changes the familiar hump-shaped

response in consumption to a monotone one. Second, more cautious households drive a

sharp initial drop in aggregate consumption. This larger drop in consumption is driven by

the large negative wealth effect associated with the higher likelihood of economic disaster

reducing TFP and earnings further. Lastly, aggregate investment experiences a large drop

44Recall that the likelihood of disaster in normal times is near zero.

37

as the lower expected future return on capital causes a strong portfolio adjustment into

liquid assets.45

As seen in Figure 8, if the stock of liquid assets is fixed, the rise in precautionary sav-

ings through liquid assets by cautious households in a recession lowers the return on liquid

savings (Figure 8 (d)). This fall in the return on liquid savings dampens the effect of lower

expected future average return on asset portfolios seen in Figure 7 as the current return on

savings also falls more than before. While weakening the change in the substitution effect

from a rise in disaster risk, the rising probability of disaster drives a far larger negative

wealth effect as it sharply reduces expected earnings compared to normal times. This drives

a severe recession from an empirically consistent actual fall in TFP.

A rise in the risk of a further worsening of earnings has little implication for aggregate

dynamics in a single asset economy. Figure 9 shows impulse responses for aggregate con-

sumption and investment when all households save using the same asset. While aggregate

consumption falls slightly more with a rise in disaster risk, on impact, we see that the

aggregate dynamics of consumption and investment are close to those with only a shock to

TFP. In a single asset economy, a rise in precautionary savings by households would lower

the return to savings in equilibrium, offsetting any aggregate effect.

45Note that I model the disaster state as an additional drop in TFP that reduces earnings and theinterest rate on savings. When I instead model disaster as a rise in capital depreciation which only affectsthe return on savings, consumption rarely falls beyond that in the response to actual TFP. It is importantto generate a rising risk of a fall in earning through an economic disaster rather than a lower return onsavings to explain a sharp drop in both aggregate consumption and capital investment. Thus, a rise indisaster risk does not increase precautionary savings if disaster reduces the return on assets relative toearnings.

38

Figure 7. Impulse response in a model with a constant return on liquid savings (rf fixed)

-5

-4

-3

-2

-1

0

1

1 11 21 31 41 51 61 71 81 91

per

cet

chan

ge

date

(b) consumption

tfp shocktfp shock + disaster risk

-25

-20

-15

-10

-5

0

5

1 11 21 31 41 51 61 71 81 91

per

cet

chan

ge

date

(c) investment

-2.5

-2

-1.5

-1

-0.5

0

1 11 21 31 41 51 61 71 81 91

per

cen

t ch

ang

e

date

(a) total factor productivity

-202468

1012141618

1 11 21 31 41 51 61 71 81 91

per

cen

t ch

ang

e

data

(d) liquid asset

Notes : Figure 7 shows aggregate dynamics in the heterogeneous wealth composition economy

following a 2.18 percent drop in TFP (black solid line) and a 2.18 percent drop in TFP alongside a

rise in disaster risk (blue dashed line). The vertical axis measures percent changes of each variable

from its simulation mean.

39

Figure 8. Impulse response in a model with a constant stock of liquid asset (B fixed)

-2.5

-2

-1.5

-1

-0.5

0

1 11 21 31 41 51 61 71 81 91

per

cen

t ch

ang

e

date

(a) total factor productivity

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1 11 21 31 41 51 61 71 81 91

per

cen

t ch

ang

e

date

(b) consumption

tfp shock

tfp shock + disaster risk

-14

-12

-10

-8

-6

-4

-2

0

2

1 11 21 31 41 51 61 71 81 91

per

cen

t ch

ang

e

date

(c) investment

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

1 11 21 31 41 51 61 71 81 91

per

cen

tage

chan

ge

date

(d) return on liquid asset

Notes : Figure 8 shows aggregate dynamics in the heterogeneous composition of wealth economy

with disaster risk in a response to a 2.18 percent drop (black solid line) and to a 2.18 percent

drop alongside with a rise in disaster risk (blue dashed line).

40

Figure 9. Impulse response in a single asset economy

-3

-2

-1

0

1

1 11 21 31 41 51 61 71 81 91

per

cen

t ch

ang

e

date

(a) consumption

tfp shocktfp shock + disaster risk -12

-10

-8

-6

-4

-2

0

2

1 11 21 31 41 51 61 71 81 91

per

cen

t ch

ang

e

date

(b) investment

Notes : Figure 9 shows aggregate dynamics in a single economy with disaster risk in a response to

a 2.18 percent drop (black solid line) and to a 2.18 percent drop alongside with a rise in disaster

risk (blue dashed line).

41

Table 11 summarizes peak-to-trough declines in both heterogeneous wealth composition

economies with a rise in disaster risk. As I have mentioned, if the real rate of return on safe

assets does not drop as much as predicted by a model with a fixed stock, the rise in disaster

risk drives a sharp substitution from illiquid capital to liquid assets, reproducing the drop

in investment and consumption seen in the data. Indeed, a portfolio choice economy with

a rigid return on liquid savings explains the 4 percent drop in consumption as well as the

20 percent drop in investment over the Great Recession.

Table 11. Peak-to-Trough declines: U.S. 2007 Recession and model

GDP I N C TFPdata 5.59 18.98 6.03 4.08 2.18two asset (B fixed) 5.71 12.83 5.58 2.82 2.18two asset (rf fixed) 5.71 19.89 5.58 3.82 2.18

Table 11 shows peak-to-trough declines between 2008q4 and 2009q2in the first row (Khan and Thomas, 2013). The second row is a modelwith two assets, where the return to liquid savings is fixed. The lastrow is a two asset model with a fixed stock of liquid asset.

The sharp simultaneous declines in investment and consumption, following a rise in the

probability of an economic disaster, is driven by dissimilar responses from households with

different levels of wealth. Wealth poor households, with little liquid savings, face the risk of

a sharp rise in the marginal utility of consumption following a further decline in earnings.

They respond with portfolio adjustment selling illiquid assets to pay off debt and build

liquid savings. The fall in investment is the result of a general reduction in savings by

wealthier households who, while roughly maintaining their share of illiquid assets, smooth

consumption by reducing overall savings. Table 12 shows the fraction of liquid assets to

total net worth over wealth quintiles in the steady state and over the first date of this

recession accompanied by a rise in disaster risk. As seen in the third row, higher risk of

economic disaster boosts precautionary savings by the poorest 20 percent of households;

their share of liquid savings rises from -127 percent to 18 percent when the return on liquid

savings is fixed.46

46While the increase in the share of liquid savings by the poorest 20 percent of households is amplified

42

Table 12. Fraction of liquid assets to total net worth over the quintile distributions, B(K+B)

B fixed steady state impact dates rf fixed steady state impact dates

t=0 t=1 t=2 t=0 t=1 t=2all 0.096 0.100 0.101 all 0.096 0.106 0.108Q1(poor) -1.27 0.26 0.34 Q1(poor) -1.27 0.18 0.33Q2 0.28 0.27 0.25 Q2 0.28 0.26 0.25Q3 0.17 0.17 0.17 Q3 0.17 0.16 0.17Q4 0.14 0.14 0.14 Q4 0.15 0.14 0.14Q5(wealthy) 0.07 0.07 0.07 Q5(wealthy) 0.07 0.07 0.08

Table 12 shows the share of liquid assets as a fraction of total wealth across wealth quintiles in the steadystate and on impact dates following a TFP shock with a rise in disaster risk.

Table 13 and 14 summarize the shares of households who actively adjust their illiquid

assets and the share of such adjustors, in each quintile, that actually disinvest illiquid

assets, respectively. Comparing the share of adjustors in the fifth quintile to that in the

first quintile, wealthy households are generally more willing to adjust their asset portfolios

as they can afford frequent transaction costs. While the share of adjustors changes little

over the business cycle, Table 14 shows that more adjusting households monetize their

illiquid assets in a recession. For instance, in the model with a rigid real return on liquid

savings, the fraction of adjustors disinvesting in illiquid assets, in the recession, increases

by 4 percentage points relative to the steady state. While poor households overall increase

their liquid assets, a higher fraction of wealth poor adjustors increase their illiquid assets

in a recession compared to the steady state as the price of illiquid assets falls.

with a fall in the rate of return on liquid savings, a rise in disaster risk still boosts their precautionarysavings compared to a recession without a rise in the probability of disaster. Indeed, without disaster risk,the start of a recession sees the share of liquid savings to net worth rise to 20 percent (fixed return model)and 13 percent (fixed stock model).

43

Table 13. Share of households who actively adjust their portfolios across wealth quintiles

B fixed steady state impact dates rf fixed steady state impact dates

t=0 t=1 t=2 t=0 t=1 t=2all 0.15 0.15 0.15 all 0.14 0.16 0.15Q1(poor) 0.06 0.07 0.08 Q1(poor) 0.06 0.07 0.08Q2 0.12 0.11 0.11 Q2 0.12 0.11 0.11Q3 0.14 0.14 0.13 Q3 0.13 0.14 0.13Q4 0.15 0.15 0.15 Q4 0.15 0.16 0.15Q5(wealthy) 0.27 0.28 0.29 Q5(wealthy) 0.26 0.29 0.28

Table 13 shows the share of households who actively adjust their illiquid assets over wealth quintiles inthe steady state and during impact dates.

Table 14. Share of adjustors in each quintile that disinvest in illiquid assets

B fixed steady state impact dates rf fixed steady state impact dates

t=0 t=1 t=2 t=0 t=1 t=2Q1(poor) 0.10 0.07 0.09 Q1(poor) 0.10 0.08 0.08Q2 0.51 0.47 0.47 Q2 0.52 0.49 0.48Q3 0.69 0.71 0.71 Q3 0.71 0.73 0.72Q4 0.79 0.83 0.83 Q4 0.82 0.85 0.83Q5(wealthy) 0.82 0.85 0.86 Q5(wealthy) 0.82 0.86 0.86

44

6.3 Changes in the joint distribution of net worth, income, and

consumption before and during the Great Recession

In this section, I examine the heterogeneous wealth composition economies for the

dynamics of net worth, income, and consumption across households, comparing to the PSID

before and during the Great Recession. I first compare changes in the joint distribution

in the data to those in heterogeneous wealth composition economies without disaster risk.

Next, I discuss how disaster risk improves the models prediction on the dynamics of the

cross-sectional distributions.

6.3.1 Recession without rising disaster risk

I simulate 60,000 households for several periods in (1) a heterogeneous composition

wealth model with a constant return to liquid savings and (2) a model with a fixed stock of

liquid assets. Table 15 compares annualized growth rates of the average level of net worth,

earnings, income, and consumption to those in the PSID between 2005 and 2007 over the

2005 wealth quintiles. In the last column of Table 15, I also report the percentage points

changes in expenditure rates defined as the fraction of total consumption to income. Note

that growth rates of variables during normal times in data are comparable to those in a

steady state in model economies.47

Table 15 shows that the model economies with multiple assets successfully explain some

stylized facts seen in the PSID before the Great Recession. First, net worth growth rates fall

sharply in the level of wealth. For example, it explains the more than 10 percent rise in net

worth for households in the second quintile compared to the small rise in net worth for the

wealthiest 20 percent of households. Second, growth rates of income and earnings in model

economies are broadly consistent with those in the PSID, generating the highest growth

rates for the bottom 20 percent of households. Falling growth rates of net worth, earnings,

and income, in the level of wealth, is mainly driven by high mean-reversion in earnings

shocks. This mean-reversion in shocks also leads to a higher growth in consumption for

47In Table 12, the minor difference in the distribution of households drives slightly different measuresbetween the two model economies.

45

the bottom compared to that for the top. However, as consumption rises less than income,

expenditure rates fall for all households but the top 20 percent.

Table 15. Growth rates of variables across wealth quintiles before the Great Recession2005-2007

Net worth Earning Income Expend. Exp. Rate (pp)

Quantile PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2)Q1(poor) n/a n/a n/a 3.3 9.8 9.3 3.3 9.8 9.9 11 7.0 6.9 3.6 -1.4 -1.5Q2 25 10.9 11.2 1.7 5.8 5.4 1.9 5.8 5.8 7.1 3.0 3.1 2.3 -1.7 -1.6Q3 22 2.9 3.0 1.8 2.5 2.4 2.0 2.5 2.5 3.0 0.8 0.9 0.5 -1.1 -1.0Q4 11 1.3 1.4 1.3 -0.1 -0.5 2.9 -0.1 -0.1 2.9 -2.1 -2.0 0.0 -1.4 -1.3Q5(wealthy) 2.1 0.2 0.2 2.6 -2.1 -2.6 1.2 -2.1 -2.1 4.3 0.2 -0.2 1.5 1.6 1.6

Notes : Table 15 shows annualized growth rates of the average net worth, earnings, and income and percentage pointschange in expenditure rates from the PSID and in portfolio choice economies without disaster risk. (1) a model with fixedreturn on liquid assets. (2) a model with fixed stock of liquid assets.

Table 16 compares the corresponding growth rates between 2007 and 2011 over the 2007

wealth quintiles in the PSID to those in the model economies following a continuous drop

in TFP, accumulating to an overall 2.18 percent across two periods. The model economies

successfully reproduce a concurrent slowdown in the growth of net worth, earnings, and

income compared to normal times (Table 15). Moreover, it explains a fall in the level of

net worth, earnings and income for those in the highest quintile of the wealth distribution.

The growth rates of the average consumption across the quintiles of the wealth distribution

decline in a recession, reproducing an actual drop in the consumption for the wealthiest 60

percent of households. The model economies also explain a rise in savings rates for house-

holds in the first four quintiles as well as a rise in consumption rate for the top 20 percent

of households. A rise in consumption rate for the top is due to the fact that wealthy house-

holds are likely to be the old, who have less desire to save as they have a shorter remaining

lifetime compared to the young.

To further evaluate the model’s predictions for the severity of the Great Recession

across the wealth distribution, Table 17 documents the first difference in growth rates

46

Table 16. Growth rates of variables across wealth quintiles during the Great Recession2007-2011

Net worth Earning Income Expend. Exp. Rate (pp)

Quantile PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2)Q1(poor) n/a n/a n/a 2.1 6.0 6.0 2.5 7.9 7.9 0.2 4.9 5.4 -2.3 -1.6 -1.3Q2 11 9.9 10.1 1.1 2.7 2.9 0.8 4.1 4.2 1.9 1.6 1.8 1.0 -1.6 -1.4Q3 -5.1 1.8 2.0 -0.3 0.1 0.2 -0.1 1.1 1.0 -0.2 -0.6 -0.2 -0.1 -1.1 -0.8Q4 -0.5 0.3 0.4 -0.7 -2.3 -2.4 -0.1 -1.4 -1.6 -1.6 -3.3 -3.0 -1.7 -1.4 -1.0Q5(wealthy) -3.5 -0.8 -0.8 0.0 -4.3 -4.2 0.3 -3.4 -3.5 -2.5 -0.8 -0.7 -2.8 1.8 2.1

Notes : Table 16 shows annualized growth rates of the average net worth, earnings, and income and percentage pointschange in expenditure rates from the PSID and in portfolio choice economies without disaster risk. (1) a model with fixedreturn on liquid assets. (2) a model with fixed stock of liquid assets.

between normal times and recession in model to the data. While not targeted, the two

model economies explain a slowdown in the growth of net worth, earnings, income, and

consumption in a recession compared to a normal time. Although the model economies

explain fall in consumption rates across the wealth distribution in a recession (Table 16),

it fails to explain changes in the growth of expenditure rates between normal times and

recession as seen in the last column of Table 17. For example, the PSID data shows that a

fall in consumption rates for households across wealth quintiles is pronounced in the Great

Recession compared to normal times while the model economies predicts a little drop or

rise in consumption rate in a recession.

Table 17. Changes in growth rates of variables between prior- and during the GreatRecession

Net Worth Earning Income Expend. Exp. Rate (pp)

Quantile PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2)Q1(poor) n/a n/a n/a -1.2 -3.8 -3.3 -0.8 -1.9 -2.0 -11 -2.1 -1.5 -5.9 -0.2 0.2Q2 -14 -1.0 -1.1 -0.6 -3.1 -2.5 -1.1 -1.7 -1.6 -5.2 -1.4 -1.3 -1.3 0.1 0.2Q3 -27 -1.1 -1.0 -2.1 -2.4 -2.2 -2.1 -1.5 -1.5 -3.2 -1.4 -1.1 -0.6 0.0 0.2Q4 -12 -1.0 -1.0 -2.0 -2.2 -1.9 -3.0 -1.3 -1.5 -4.5 -1.2 -1.0 -1.7 0.0 0.3Q5(wealthy) -6 -1.0 -1.0 -2.6 -2.2 -1.6 -0.9 -1.3 -1.4 -6.8 -1.0 -0.9 -4.3 0.2 0.5

Notes : Table 17 shows changes in annualized growth rates of the average net worth, earnings, and income and percentagepoints change in expenditure rates from the PSID and in portfolio choice economies without disaster risk. (1) a modelwith fixed return on liquid assets. (2) a model with fixed stock of liquid assets.

47

6.3.2 With disaster risk

Table 18 reports the annualized growth rates of net worth, earnings, income and con-

sumption in the 2007-2010 PSID as well as from the simulated heterogeneous wealth com-

position economies with disaster risk.48 While growth rates in net worth, earnings, and

income are broadly similar to those in Table 16, the slowdown in consumption growth in

a recession is exacerbated by a rise in disaster risk, especially when the return to liquid

assets is fixed. For example, in the model with fixed returns, consumption for the bottom

20 percent of households rises by 4.1 percent in a recession with a rise in disaster risk com-

pared to 4.9 percent without disaster risk (Table 16). As explained earlier, this is driven by

a higher expected return on savings in the future compared to the current average return

alongside with a strong negative wealth effect following a rise in disaster risk.

Table 18. Growth rates of variables across wealth quintiles during the Great Recessionwith rising disaster risk

Net worth Earning Income Expend. Exp. Rate (pp)

Quantile PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2)Q1(poor) n/a n/a n/a 2.1 7.3 7.4 2.5 7.8 7.8 0.2 4.1 5.4 -2.3 -2.0 -1.2Q2 11 9.7 10.6 1.1 3.7 3.9 0.8 4.3 4.0 1.9 0.9 1.9 1.0 -2.1 -1.3Q3 -5.1 1.8 2.2 -0.3 1.0 1.0 -0.1 1.2 0.7 -0.2 -1.0 -0.1 -0.1 -1.5 -0.5Q4 -0.5 -0.1 0.5 -0.7 -1.7 -1.6 -0.1 -1.1 -1.8 -1.6 -3.8 -2.9 -1.7 -1.9 -0.8Q5(wealthy) -3.5 -1.1 -0.8 0.0 -3.9 -3.8 0.3 -3.0 -3.7 -2.5 -1.1 -0.7 -2.8 1.4 2.2

Notes : Table 18 shows annualized growth rates of the average net worth, earnings, and income and percentagepoints change in expenditure rates from the PSID and in portfolio choice economies with disaster risk. (1) a modelwith fixed return on liquid assets. (2) a model with fixed stock of liquid assets.

Table 19 shows the annualized changes in the joint distribution between normal times

and a recession. Involving a rise in disaster risk when the return on liquid savings is con-

stant, this model economy predicts a more marked fall in consumption rates. This implies

that a rise in disaster risk not only explains the aggregate response of the economy seen in

the data but also the micro-prediction of the economy. In a typical recession with a single

asset, households’ consumption smoothing desire increases consumption rates. However,

48Note that the growth rates in normal times are same as those reported in Table 13.

48

in the recent recession, consumption rates sharply fall across households, which can be ex-

plained with a rise in the probability of a further worsening of earnings with a rigid return

on liquid savings. This suggests the importance of disaster risk that amplifies the effects

of a shock to TFP with multiple assets of varying liquidity.

Table 19. Changes in growth rates of variables between prior- and during the GreatRecession with rising disaster risk

Net Worth earning Income Expend. Exp. Rate (pp)

Quantile PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2)Q1(poor) n/a n/a n/a -1.2 -2.5 -1.9 -0.8 -2.0 -2.1 -11 -2.9 -1.5 -5.9 -0.6 0.3Q2 -14 -1.2 -0.6 -0.6 -2.1 -1.5 -1.1 -1.5 -1.8 -5.2 -2.1 -1.2 -1.3 -0.4 0.3Q3 -27 -1.1 -0.8 -2.1 -1.5 -1.4 -2.1 -1.3 -1.8 -3.2 -1.8 -1.0 -0.6 -0.4 0.5Q4 -12 -1.4 -0.9 -2.0 -1.6 -1.1 -3.0 -1.0 -1.7 -4.5 -1.7 -0.9 -1.7 -0.5 0.5Q5(wealthy) -6 -1.3 -1.0 -2.6 -1.8 -1.2 -0.9 -0.9 -1.6 -6.8 -1.3 -0.5 -4.3 -0.2 0.6

Notes : Table 19 shows changes in annualized growth rates of the average net worth, earnings, and income and percentagepoints change in expenditure rates from the PSID and in portfolio choice economies with disaster risk. (1) a model withfixed return on liquid assets. (2) a model with fixed stock of liquid assets.

49

7 Concluding Remarks

Solving an overlapping generations economy with uninsurable earnings, unemployment

and liquidity risk in dynamic stochastic general equilibrium, I have examined the impli-

cations of household-level differences in the level and liquidity of wealth for aggregate

dynamics and changes in the distribution of consumption, income and wealth across house-

holds. In particular, I quantify the role of household heterogeneity in the composition of

wealth for understanding differences in their responses during the Great Recession. This

channel is amplified when households increase precautionary savings, and substitute liquid

for illiquid assets, following a rise in disaster risk.

Liquidity varies across assets as higher returns involve investments associated with ran-

dom transactions cost. Calibrating the distribution of liquidity costs to the real return on

liquid and illiquid assets in the data, the model economy reproduces the empirical share of

illiquid assets to net worth of households of different levels of wealth and age. Moreover,

the model explains much of the distribution in net worth, illiquid wealth, and liquid wealth

seen in the data.

The implications of adding realistic heterogeneity not only in the level of wealth, but also

in its liquidity, are important for an understanding of both aggregate and household-level

changes over large recessions. In such recessions, a fall in earnings and a rise in unemploy-

ment risk force high-yield asset investors to monetize their wealth to smooth consumption,

resulting in a sharp drop in investment. During the recovery, as fixed transaction costs

lead households to tolerate lower than desired shares of illiquid assets in their portfolios,

the economy with wealth composition heterogeneity recovers slowly. Most importantly,

precautionary savings motives, following an increase in the probability of a large disaster,

yield powerful changes in household behavior. These changes in households’ consumption

and expenditure rates are consistent with stylized facts seen in the joint distribution of

consumption, income and net worth across households over the Great Recession.

In the next version of this paper, I will use the empirical cyclicality of liquidity premia

to determine the cyclicality of the supply of liquid assets over the business cycle. This

will provide further information on the role of household portfolio substitutions, associated

50

with an increase in precautionary savings, in large recessions.

51

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A Data appendix

A.1 2007-2009 SCF panel data

The SCF is household triennial data survey conducted by the Federal Reserve Board.

The SCF provides detailed information on assets and debts of households. The SCF also

employs a list sample design based on the IRS data to provide disproportionate sampling of

wealthy households. Historically, the SCF had panel data from 1983 to 1989 and recently

released two-year panel data for 2007 and 2009.

Net worth is defined as total assets minus total debt. Total assets include both financial

and non-financial assets. Total financial assets include transaction accounts, certificates of

deposit, directly held pooled investment funds, savings bonds, directly held stocks and

bonds, cash value of life insurance, quasi-liquid retirement accounts, and other managed

and miscellaneous financial assets. Total non-financial assets include vehicles, residential

property, net equity in non-residential real estate, business equity, and other miscellaneous

non-financial assets. Total debt consists of debt secured by residential property, credit card

balances since last payment, installment loans, and other debts. I select households with

a head who is between 25 and 85 years old and not self-employed. All the variables are

deflated and expressed in 2013 dollars. Full sample weights are used.

A.2 PSID data

The PSID is a longitudinal survey of a sample of US individuals and families conducted

annually from 1968 to 1977, and biennially since 1997. The original 1968 PSID sample

combines the Survey Research Center (SRC) and the Survey of Economic Opportunities

(SEO) samples. I use the U.S population representative SRC sample.

For greater consistency with the SCF, I select households with a head who is between

25 and 85 years old and drop the self-employed. I drop a sample observation if income is

positive but annual hours worked is zero. If income is top-coded, I multiply it by a factor

of 1.5 of the top-coded threshold following Katz et al. (1999). All variables are deflated

and expressed in 2013 dollars using IPUMS-CPI.

55

The PSID provides disaggregated data on wealth since 2003, consisting of business and

farm equity, transaction accounts, equity in real estate and vehicles, stocks, bonds, IRA,

and debt. As shown in Table 1, the distributions of wealth in the PSID are comparable to

those in the SCF. Table A1 also summarizes averages in the PSID and SCF.

Table A1. Summary statistics of data

Data SCF PSIDyear 2007 2009 2007 2009

Sample size 13,085 13,085 2,524 2,524Age of head 49 51 42 44

Total wealth 395,750 325,060 344,161 283,959

Head income 75,336 71,949 72,365 72,690

Notes : All variables are expressed in 2013 dollars. For the PSID,drop three samples with wealth less than negative 99 million dol-lars. (Source: 2007-2009 PSID and SCF)

The PSID provides data on households’ consumption. In addition to food and housing,

the PSID included items on transportation, health care, education, utilities, and child care

since 1999. In 2005, additional items such as household furnishing and equipment, clothing

and apparel, trips and vacations, and recreation and entertainment were added.

I construct total expenditure in the PSID as the sum of nondurable goods and ser-

vices.49 The PSID measures total spending on each item for the family. Since each item

has a different reporting time unit, I adjust it to an annual measure. Reporting time unit

varies by samples for food delivered, food eaten out, and food at home. Approximately

half of respondents reported it weekly while the rest report it monthly or biweekly. I mea-

sure annual spending on these items based on individual reported time units. Following

Krueger et al. (2016), I imputed the amount of rental services from home owners by mul-

tiplying the value of the main residence by 4 percent. Imputed rent and property taxes

are included in expenditure on housing to be consistent with the BEA measure. Given

that reported income is earned in the preceding year, there may exist time inconsistency

49Durable goods include motor vehicles and parts, furnishings and equipment, recreational goods andvehicles, and other durable goods.

56

between consumption and income.

In Table A2 and A3, I compare composition of total expenditure in the PSID to that

in the BEA to comprehend the representativeness of the former micro data for macro ag-

gregates.50 I used NIPA Table 2.3.6 Real Personal Consumption Expenditures in chain

2009 dollars (seasonally adjusted).51 Since the BEA measures current year consumption

while the PSID reporting time unit varies by item, I take the average of the current and

preceding years of the expenditure in the BEA to make the time unit closer to the expen-

diture measured in the PSID. Moreover, I divide total expenditure in PSID by total family

size and multiply the consumption per capita by the population size to make household

consumption in PSID comparable to Personal Consumption Expenditure in BEA, Table

A2 shows that the PSID aggregate consumption accounts for more than 50 percent of total

expenditure in the BEA. In Table A3, spending on each item as a share of total expenditure

in the PSID is broadly comparable to that in the BEA. This suggests that, while micro data

captures less expenditure than aggregate data in total, the dynamics in consumption at

individual level may be explained by micro data. For instance, Krueger et al. (2016) shows

that the growth rates of the PSID aggregate closely follow BEA total spending, providing

further support for using micro consumption data to measure individual growth rates.

50Total spending in the recent PSID is comparable to that in the CEX. Indeed, Charles et al. (2007)find that the 2003 PSID covers 72 percent of total expenditure measured in CEX.

51Given the issue raised by chain-weighted dollars, I express PSID variables in 2009 dollars for TableA2 and A3.

57

Table A2. Composition of total expenditure in the BEA and PSID

2005 2007 2009billions of 2009 dollars BEA PSID BEA PSID BEA PSID

Nondurable goods 2,098 1,397 2,208 1,497 2,243 1,350food 744 900 786 950 776 916clothing 299 220 321 207 314 181gasoline 299 278 297 340 284 253other 753 n/a 814 n/a 821 n/a

Services 6,255 3,039 6,592 3,555 6,679 3,404housing and utilities 1,753 1,371 1,832 1,843 1,871 1,689health care 1,466 300 1,544 345 1,613 366transportation 333 280 337 296 306 285recreation 360 122 385 123 383 126food services 595 n/a 629 n/a 613 n/afinancial services and insurance 686 265 731 238 728 199other services 856 701 895 711 891 739

total 8,353 4,436 8,812 5,052 8,873 4,754

Notes : Variables in the BEA are expressed in chain-weighted 2009 dollars. Variables in the PSIDare expressed in 2009 dollars. Other services include chicle care, education, communication andvehicle services.

Table A3. Spending as a fraction of total expenditure in the BEA and PSID

2005 2007 2009% of total expenditure BEA PSID BEA PSID BEA PSIDNondurable goods 25.1 31.5 25.1 29.6 25.3 28.4

food 8.9 20.3 8.9 18.8 8.8 19.3clothing 3.6 5.0 3.6 4.1 3.5 3.8gasoline 3.6 6.3 3.4 6.7 3.2 5.3other 9.0 n/a 9.2 n/a 9.3 n/a

Services 74.9 68.5 74.8 70.4 75.3 71.6housing and utilities 21.0 30.9 20.8 36.5 21.1 35.5health care 17.6 6.8 17.5 6.8 18.2 7.7transportation 4.0 6.3 3.8 5.9 3.5 6.0recreation 4.3 2.8 4.4 2.4 4.3 2.7food services 7.1 n/a 7.1 n/a 6.9 n/afinancial services and insurance 8.2 6.0 8.3 4.7 8.2 4.2other services 10.3 15.8 10.2 14.1 10.0 15.5

total 100 100 100 100 100 100

Notes : Variables in the BEA are expressed in chain-weighted 2009 dollars. Variables in the PSIDare expressed in 2009 dollars. Other services include chicle care, education, communication andvehicle services.

58

Figure A1. Illiquid wealth share and risky wealth growth rate over 2007 wealth deciles

-50

0

50

100

150

200

250

0

0.5

1

1.5

2

2.5

3

3.5

4

2 3 4 5 6 7 8 9 10

illi

qu

id w

ealt

h g

row

th r

ates

illi

qu

id w

ealt

h s

har

e 0

7

2007 wealth deciles

illiquid wealth share 07 (as a fraction of asset) illiquid wealth share 07 (as a fraction of wealth)

illiquid wealth growth rates

Notes : The average illiquid asset share as a fraction of total asset (blue bars) and as a fraction of

net worth (grey bars) in 2007. This measure only considers samples who invest in illiquid wealth.

Exclude samples with negative net worth (asset) in 2007 for net worth (asset) share measures or

with net worth less than 0.1 percent of the average level of wealth. The growth rates of illiquid

wealth between 2007 and 2009 (dashed line). Exclude samples with negative values in any sample

period for growth rate measures. (Source: 2007-2009 SCF panel)

59

Figure A2. Illiquid wealth share and risky wealth growth rate over 2007 age groups

-40

-35

-30

-25

-20

-15

-10

-5

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

25-34 35-44 45-54 55-64 65-74 over 75

illi

qu

id w

ealt

h g

row

h r

ates

illi

qu

id w

ealt

h s

har

e 0

7

2007 age bin

illiquid wealth share 07 (as a fraction of asset) illiquid wealth share 07 (as a fraction of wealth)

illiquid wealth growth rates

Notes : The average illiquid asset share as a fraction of total asset (blue bars) and as a fraction of

net worth (grey bars) in 2007. This measure only considers samples who invest in illiquid wealth.

Exclude samples with negative net worth (asset) in 2007 for net worth (asset) share measures or

with net worth less than 0.1 percent of the average level of wealth. The growth rates of illiquid

wealth between 2007 and 2009 (dashed line). Exclude samples with negative values in any sample

period for growth rate measures. (Source: 2007-2009 SCF panel)

60

Figure A3. The composition of illiquid wealth in 2007

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 3 4 5 6 7 8 9 10

2007 wealth deciles

stocks business equity house residential property non-residential vehicle

Notes : The average share of each illiquid wealth as a fraction of total illiquid wealth. Exclude

samples with negative or zero risky wealth. (Source: 2007-2009 SCF panel)

61

B Competitive Investment Firm

A competitive investment firm has the technology that creates capital. It holds k units

of capital created last period after selling p(z, µ)k in shares to households. An investment

firm rents this capital to a production firm at the rental rate rk(zf , µ) and returns the

current dividend value of shares to households. It also faces convex capital adjustment cost

Φ(k′, k). A competitive investment firm chooses k′ to maximize its profit

J(k, zf , µ) = maxk′

((rk(zf , µ) + 1− δ)k − (p(zf , µ) + d(zf , µ))k

+ p(zf , µ)k′ − k′ − Φ(k′, k) +

nz∑g=1

πfgr(zg, zf , µ)J(k′, zg, µ

′)

)(4)

where investment firm discounts future earnings by the marginal rate of substitution of

households, r(zg, zf , µ).

The investment firm’s optimal choices satisfy the following first-order condition

p(zf , µ)− 1− Φ1(k′, k) +

nz∑g=1

πfgr(zg, zf , µ)D2J(k′, zg, µ

′) = 0 (5)

and the Benvensite-Scheinkman condition provides

D2J(zf , k, µ) = rk(zf , µ) + 1− δ − (p(zf , µ) + d(zf , µ))− Φ2(k′, k). (6)

Assuming perfect competition for an investment firm, zero profit condition,D2J(k′, zg, µ

′) =

0, determines the equilibrium price of capital and dividends as follows.

p(zf , µ) = 1 + Φ1(k′, k) (7)

d(zf , µ) = rk(zf , µ)− δ − Φ1(k′, k)− Φ2(k

′, k) (8)

62

C Equilibrium Ex-dividend Price and Dividends

In this section, I show that definitions of ex-dividend price and dividend are consistent

with equilibrium and imply the aggregate resource constraint. Here, I simplify the model

by abstracting from government and liquid asset market.

Aggregate budget constraint for all households who are adjusting illiquid wealth is as

follows:

ca + p(z, µ)k′a ≤ wa(z, µ) + (p(z, µ) + d(z, µ))ka − ζa

where xa is the sum of a variable x for all households who are adjusting and wa(z, µ) is the

labor income for all active adjustors.

Using definitions of ex-dividend price and dividend, above aggregate budget constraint

can be re-written as:

ca + (1 + Φ1(k′, k))k′

a ≤ wa(z, µ) + (αηkα−1n1−α − δ − Φ2(k′, k))ka − ζa (9)

where k is the aggregate stock of capital.

Likewise, aggregate budget constraint for all households who are non-adjusting illiquid

wealth is:

cn ≤ wn(z, µ) (10)

k′n = (1 + αηkα−1n1−α − δ − Φ1(k

′, k)− Φ2(k′, k))kn (11)

Imposing xa + xb = x and Φ1(k′, k)k′ + Φ2(k

′, k)k = Φ(k′, k), equations (4)-(6) imply

the aggregate resource constraint

c+ k′ + Φ(k′, k) ≤ y + (1− δ)k − ζa

63

D Numerical Method

D.1 Steady state

Stationary equilibria involve finite horizon dynamic programming problems with two

endogenous state variables - illiquid wealth, a, and liquid wealth, b. The state space

is six-dimensional: age, illiquid wealth, liquid wealth, unemployment shock, persistent

and transitory earnings shocks. Solving the model with two endogenous state variables

involves a significant amount of memory (RAM) to store decision rules as well as a lot of

computation time. In solving the model, I combine unemployment shock, persistent and

transitory shock processes into a single shock process ε, decreasing two dimension. More

importantly, I develop a two-stage approach to solve savings decisions with two assets. In

the first stage, given a current fixed cost, a household chooses whether or not to adjust

its portfolio. If it adjusts, it chooses its savings in illiquid wealth, a′. In the second stage,

given a′, I solve for the optimal choice of liquid wealth, b′, using endogenous grid method

(Carroll, 2006).52

As the aggregate supply of liquid wealth is calibrated to match a rate of return of zero

percent on liquid wealth, solving for stationary equilibria involves three prices (w, p, d).

Note that, in the absence of endogenous labor supply, wages are determined by the aggre-

gate stock of capital. Moreover, p is fixed to one given equilibrium price function as there

is no aggregate capital adjustment cost in a steady state. Given the initial guess of prices,

I compute decision rules and the distribution of households. The latter is determined using

a large grid over age, idiosyncratic shock, illiquid and liquid wealth. I use bilinear interpo-

lation to place decision rules onto this grid. I update prices by bisecting for the aggregate

capital, iterating through the above steps, until prices converge.

D.2 Decision rules

I solve the household’s problem in two stages. Here, I abstract from aggregate states for

the ease of notation. In the first stage, it has decided its illiquid wealth for the next period

52The prime denotes variables in the next period.

64

, a′. This may be (1+ (1− τa)d)a if a household chooses not to adjust, or it is the result of

an active portfolio adjustment choice for a′ after paying the fixed cost ζ. Below, I describe

the household’s problem at age j with illiquid asset a, liquid wealth b, and productivity

(working status) ε. Define v0j as the intermediate value defined over cash-on-hand, m, the

future stock of illiquid wealth a′, and current productivity.

The illiquid wealth problem

vj (a, b, εi, ζ) = max{

max0≤a′≤m

v0j (m− pa′, a′, εi) , v0j (xi + b, (1 + (1− τa)d) a, εi)

}(12)

subject to

m = xi(j, ε) + (p+ (1− τa)d) a+ b− ζ

Note that if a household chooses to adjust illiquid wealth to a′, the remaining cash-

on-hand for consumption and liquid wealth in the second stage is m − pa′. However, it is

able to cash in its current stock of illiquid wealth, a. If a household does not pay its fixed

cost, it can not adjust the current stock of illiquid wealth and the cash available for liquid

wealth and consumption is the sum of labor income if working or pension benefit if retired

and the current stock of liquid wealth.

In the second stage, a household has already decided its illiquid wealth for the next

period, a′. Given a′ and remaining cash-on-hand m, a household solves the problem below.

The consumption and liquid wealth problem

v0j (m, a′, εi) = maxb′

(u (c) + βvej (a

′, b′, εi))

(13)

subject to

c+ qb′ ≤ m

65

where vej represents the expected value of a household at the beginning of the next period

before fixed cost is drawn.

D.3 Aggregate Dynamics

To solve the model with aggregate uncertainty, I extend the Backward induction method

of Reiter (2002, 2010) to solve a stochastic overlapping-generations economy. This involves

generalizing the method to handle bivariate cross-sectional distributions of endogenous state

variables. The Backward induction method of Reiter allows the distribution of households

to vary in potentially rich ways as a function of an approximate aggregate state as it does

not impose a parametric aggregate law of motion. Moreover, this method does not involve

repeated simulation, reducing computation time compared to Krusell and Smith (1998).5354

Backward induction method of Reiter (2002, 2010) selects a proxy distribution across a

grid of approximate aggregate state based on distribution selection function (DSF) which

maps approximate aggregate states to cross-sectional distributions. A DSF selects the

proxy distribution that minimizes the distance to the reference distribution subject to mo-

ment consistency conditions.55 Solving for the DSF entails solving a large system of linear

equations. With proxy distributions solved, backward induction simultaneously solves for

households’ decision rules and an end-of-period distribution implying a future approximate

aggregate state consistent with households’ expectations. This enforces consistency be-

tween individual behavior and the aggregate law of motion. Lastly, I simulate the model

economy and weight simulated distributions using an inverse quadratic to update the ref-

erence distributions, and thus the DSF.

This paper contributes to the backward induction method of Reiter (2002, 2010) in

two ways. First, I solve a stochastic OLG economy involving distributions defined over

53Both Krusell and Smith (1998) and Reiter (2002, 2010) solve households’ decisions over approximateaggregate states which summarizes infinite-dimensional cross-sectional distributions using a finite vector ofmoments.

54Krusell and Smith (1998) assume a parametric function to forecast an approximate aggregate state.Though Krusell and Smith (1998) update forecast rules based on realistic simulation-generated distribu-tions, it is critical to have a long simulation to avoid sampling errors. These repeated long period simulationsmake it costly to solve a model with rich distribution of households using the Krusell and Smith’s method.

55I initially use the steady state distribution as a reference distribution.

66

60 different cohorts. The distribution of households over age and idiosyncratic shocks as

well as two assets increases the dimension of the system of equations solved for proxy

distributions making it intractable. To mitigate this problem, I aggregate full reference

distributions over age and idiosyncratic shocks into a small subset of age and idiosyncratic

type groups and calculate the weights mapping full distributions to aggregated ones before

solving for proxy distributions. These weights keep the shape of proxy distributions over age

and idiosyncratic shocks conditional on the level of wealth close to that of the original full

distributions. Second, I solve the model with two endogenous state variables - illiquid and

liquid wealth. To make the solution feasible, I solve the model with aggregate uncertainty

over asset grids with a lower number of grid points than used for the steady state, both for

decision rules and distributions. Having finished solving this model, I simulate the model

economy over finer grids to have more accurate solution for households’ decisions.

I summarize the outline of the algorithm as follows:

(1) Approximate the cross-sectional distribution in the aggregate state, z = {z1, ..., znz},

with a finite vector of statistics (moments) m = {m1, ...,mnm}. Here, I assume mi is ith-

moment.

(2) Determine asset grids for decision rules and distributions with a lower number of grid

points, A = {a1, .., ana} and B = {b1, .., bnb} , keeping bounds the same as those in the

steady state. These grids are used until the backward induction is solved.

(3) Aggregate the full reference distribution, rµ(j, a, b, ε; z,m), across all age groups and a

small subset of idiosyncratic types nε ≤ nε, resulting in the reduced distribution, rµ0 (a, b, ε; z,m)

where ε ∈ {ε1, ..., εnε}, and calculate weights of this mapping.

ω0(j, a, b, ε, ε; z,m) : rµ0 (a, b, ε; z,m) → rµ(j, a, b, ε; z,m)

(4) Choose a DSF which gives the proxy distribution, pµ0(a, b, ε; z,m). I solve for pµ0(a, b, ε; z,m)

as the solution to a problem that minimizes the distance to the reduced distribution

rµ0 (a, b, ε; z,m) while imposing that each ε sums to its reference value and moment consis-

tency constraints. For each approximate aggregate state (z,m), a DSF solves :

67

min{pµ0 (ai,bk,εl)} na, nb, nε

i=1,k=1,l=1

na∑i=1

nb∑k=1

nε∑l=1

(pµ0(ai, bk, εl)− rµ0 (ai, bk, εl))2

subject to

na∑i=1

nb∑k=1

pµ0(ai, bk, εl) =na∑i=1

nb∑k=1

rµ0 (ai, bk, εl), l = 1, . . . , nε (14)

na∑i=1

nb∑k=1

nε∑l=1

pµ0(ai, bk, εl)aimi = ma

im , im = 1, . . . , nm (15)

na∑i=1

nb∑k=1

nε∑l=1

pµ0(ai, bk, εl)bimk = mb

im , im = 1, . . . , nm (16)

pµ0(ai, bk, εl) ≥ 0, ∀ i, k, l

where equation (14) represents type consistency conditions. Equations (15) and (16) are

moment consistency constraints for both assets. Lastly, probabilities should be positive.

Ignoring non-negativity constraints for probabilities, the first-order condition for pµ0(ai, bk, εl),

with λl, λaim , and λb

im as Lagrange multipliers for (14), (15), and (16) respectively, is

2(pµ0(ai, bk, εl)− rµ0 (ai, bk, εl))− λl −nm∑

im=1

λaima

imi −

nm∑im=1

λbimb

imk = 0 (17)

Finally, I solve a system of nεnanb+nε+2nm linear equations in ({pµ0(ai, bk, εl)}na, nb, nε

i=1,k=1,l=1,

{λl}nεl=1, {λa

im}nmim=1, {λb

im}nmim=1).

As I ignored non-negativity constraints, the resulting solution to the system of equations

may have negative elements. If any of elements of solution are negative, I set those elements

equal to zero and reduce the system to the remaining elements. I solve the reduced system

68

iteratively until the solution has no negative elements.

(5) Using weights in (3), restore the full proxy distribution over age and idiosyncratic

shocks, pµ(j, a, b, ε; z,m).

(6) Simultaneously solve for households’ decision rules and an intrameporally consistent

future approximate aggregate m′. Guess the aggregate law of motion Gk(z,m) for approx-

imate aggregate states. Given v(J + 1, a, b, ε; z,m) = 0, solve for decision rules and value

functions backwards by age over aggregate states. Compute the full proxy distribution

consistent end-of-period aggregate state m′ and update Gk(z,m). Iterate until Gk(z,m)

converges. Note that this solves for an aggregate law of motion alongside households’ value

functions.

(7) Given the value function solved by backward induction, simulate the model economy for

T periods, then drop the first T0 periods to develop new reference distributions. The simu-

lation bisects for m′. Let µt(j, a, b, ε) be the distribution of households over the simulation

period t = T0 + 1, ..., T .

I create new reference distributions as a weighted sum of µt, putting higher weights for

distributions generating moments, mt, closer to the vector of moments m. Define index

sets that group dates for the same exogenous aggregate state, z, I(z) = {t|zt = z} where

z = z1, ..., znz . Let N(z) be the length of the vector I(z). The reference distribution for

each (z,m) is

rµ(j, a, b, ε; z,m) =1

N(z)

∑t∈I(z)

δ1(m,mt)

δ(z,m)µt(j, a, b, ε)

where δ1(m0,m1) is defined as the inverse of the Euclidian norm and δ(z,m) =∑

t∈I(z) δ1(m,mt).

(8) Iterate (3)-(7) to improve a DSF until no additional accuracy is achieved.

69

E Additional Tables

Table D1. Share of earnings, income, expenditure and expenditure rates over wealthquantiles during normal times

% share of % Expenditure rateEarnings Disp income Expend. Earnings Disp income

Quantile data model data model data model data model data modelQ1 13.8 7.1 13.4 6.3 14.7 5.2 57.6 63.5 56.9 83.3Q2 16.4 11.7 16.3 11.1 14.9 10.2 49.1 72.1 47.7 90.9Q3 19.0 13.6 18.7 12.9 17.9 12.2 51.0 79.7 49.9 96.1Q4 21.6 15.8 21.3 16.1 22.1 16.1 55.2 87.3 53.9 99.8Q5 29.1 51.7 30.3 53.7 30.2 56.4 56.1 93.1 51.9 102

Table D2. Single asset economy

x = Y C I K Bs N E(r) rf wmean(x) 2.66 2.14 0.54 7.81 n/a 1.44 0.05 n/a 1.18σx/σy (2.75) 0.35 1.94 0.21 n/a 0.86 0.92 n/a 0.29

corr(x, y) 1.0 0.94 0.99 -0.05 n/a 0.96 0.95 n/a 0.59

Notes : Table D2 presents means of GDP, consumption, investment in illiquid wealth,stock of capital, supply of liquid wealth, total hours worked, expected return onilliquid wealth, return on liquid savings and wage for the model simulated data. Italso lists relative standard deviation to and correlation with GDP for each variable.I smooth series using a HP-filter with a smoothing parameter of 100.

70

Table D3. portfolio choice economy with disaster risk

x = Y C I K Bs N p q wmean(x) 2.51 1.85 0.49 6.53 0.65 1.44 1.0 0.0 1.10σx/σy (2.56) 0.39 1.99 0.23 0.27 0.83 0.27 0.0 0.30

corr(x, y) 1.0 0.97 0.98 0.02 -0.48 0.96 0.96 0.02 0.63

Notes : Table 6 presents means of GDP, consumption, investment in illiquid wealth,stock of capital, supply of liquid wealth, total hours worked, price of illiquid wealth,and wage for the model simulated data. It also lists relative standard deviation toand correlation with GDP for each variable. I smooth series using a HP-filter with asmoothing parameter of 100.

71